This reminds me of the debate my friend Dr. Gregg Frazer and co-blogger Kristo Miettinen are having on whether Thomas Jefferson qualifies as a "Protestant Christian." Jefferson did think of himself as one. And I think both learned debaters agree not only the basics of their personal theology (orthodox Trinitarian Christianity) but also on what it was that Jefferson believed in (that he fervently denied the Trinity and believed only parts of the Bible were inspired). The question centers more around semantics: Is this -- what Jefferson believed -- just another form of "Protestant Christianity," albeit a highly "unorthodox" one?
Unmoored from church authority over official doctrine, such "unorthodox Protestant Christianity" leaves the individual and voluntary groups free 1) to determine such matters, including which parts of the biblical canon are valid, and 2) perhaps to ADD additional revelation to the Bible. Here is how Nathan Hatch, president of Wake Forest University and one of the premier scholars of Religion and the Founding Era, describes the theology of Charles Chauncy, one of the Founders unorthodox Protestant Christianity and a key theological influence on the American Founding:
Charles Chauncy, pastor of Boston’s First Church for sixty years (1727-1787), is the most prominent example of an exclusive appeal to Biblical authority in order to unravel theological orthodoxy. Chauncy was persuaded to emphasize Bible study by reading the works of English divines, such as Samuel Clarke’s The Scripture-Doctrine of the Trinity (London, 1712) and John Taylor’s The Scripture-Doctrine of Original Sin (London, 1740). Both authors used a “free, impartial and diligent” method of examining Scripture to JETTISON, respectively, the doctrines of the Trinity and of Original Sin. 
During the 1750s, after the Great Awakening, Charles Chauncy spent seven years engaged in the approach to Bible study expounded by these English authors. In the spring of 1754 he wrote to a friend,
“I have made the Scriptures my sole study for about two years; and I think I have attained to a clearer understanding of them than I ever had before.”
His studies led him to draft a lengthy manuscript in which he REJECTED the idea of eternal punishment and embraced universalism.
On to modern day Christian Nationalism and folks who might be sympathetic to the idea of an "implicit Protestant Establishment." My co-blogger Brad Hart notes how many Mormons embrace the idea of a "Christian Nation," with a quotation from a prominent member of the LDS:
As Latter-day Saints and their families study the American constitutional system, one problem they encounter is that the most popular and widely respected sources present constitutional philosophy very differently from the intent of the founders. Fortunately, the Lord has anticipated this problem and has provided the correct standard to know in which direction to strive regardless of how popular or plausible a contrary direction may be made to appear. This He has done by declaring that He established the Constitution through wise men He raised up for that very purpose. 
So the Mormon says to the evangelical and the Roman Catholic, "yes it was 'our' Jesus Christ, the same Jesus, Mormons, evangelicals and RCs believe in, who 'founded' America." To which the evangelical and Catholic obviously reply, "what do you mean by 'our' kemosabe?"
And indeed the Mormon Jesus is a more authentic "Founder" of America, given Mormonism was formed post America, and their official theology legitimately incorporates the divine inspiration of at least some of the events of the American Founding.
When orthodox Christians claim to believe in the divine claims stated in the Declaration of Independence -- ideas that although attached to a monotheistic God have nothing to do with the Bible's text -- arguably they "Mormonize" their faith by adding divine ideas to Christianity that are not found within the biblical canon. Yet orthodox Christendom has been doing this ever since it embraced natural law theology (or "natural religion"), which likewise adds divine ideas discovered from reason as a supplement to the Bible.
Or take another expositor of the Christian America idea (as my co-blogger Brad Hart cites her) Elizabeth Clare Prophet who wrote:
America is a land infused with sacred fire; America is born of God’s desire. I ask you then to secure the famous painting…of George Washington kneeling in prayer. I ask that this shall be a sign of those who love America in Christ, in God, in freedom. I ask that you give this painting to your friends who are Christians, who are religious, who are devotees, that you ask them to have it in their homes, and that you ask them to pray with you for the light and victory of America. 
Now, of course, an evangelical believer in the "Christian America" thesis would, without knowing more about Ms. Prophet, think "what a wonderful lady, I'd love to use her materials to homeschool my children." That is until they find out she believes in practically EVERY SINGLE world religion. But she's no liberal fuzzy Unitarian Universalist, but extreme right wing. She also believes Jesus Christ was an "Ascended Master," and that EVERY human being has the potential to become one (as her late husband Mark Prophet now is).
And that means humans eventually become God. Not "gods," but like some Hindus (by the way she believes in Hinduism as well) and almost a parody of Trinitarian logic, many different "personalities" of one God. Instead of just three distinct personalities of "one God," there are limitless potential distinct personalities of "one God." Her husband, Mark Prophet is now God, just as Jesus Christ is God.
As she speaks of her late husband, now the Ascended Master "Lanello."
"Looking at Mark Prophet's past lives, we see that they span the many cultures and religions of the world. Think about it. He was Noah, Lot, Ikhnaton, Aesop, the disciple Mark, Origen, Lancelot, Bodhidharma, Clovis, Saladin, Bonaventure, Louis XIV, Longfellow and the Russian czarevitch Alexis Nikolayevich."
She calls herself, among other things, a "Christian" and claims to follow a very "Christ" centered teaching. If you look at the figures she claims her husband was in past lives, the overwhelming majority of them were "biblical." I wonder if she qualifies as a "Christian" for "historical" purposes.
[Btw, her son, Sean Prophet is now an atheist and a friend of Positive Liberty. You can read his website here.]
But the Mormons and Elizabeth Clare Prophet, by the conscious design of America's Founders, get their rightful place at the table of America "implicit Protestant establishment," as just another eccentric "Protestant sect."
I claim this in part because I don't see Mormonism or Ms. Prophet's "eccentric" teachings as that much further removed from "historic Christianity" than for instance Swedenborgianism. So what do they believe?
That there is one God and that He is the Lord Jesus Christ. Within the single Person of God there is a Divine Trinity.
So off the bat, they are unorthodox in their Christology, not uncommon during America' Founding era. Though they, interestingly, are neither unitarians nor trinitarians. They also have an odd view of the "atonement" (they reject it as "satisfaction") and, like the unitarians, put more emphasis on works for salvation.
But here is where we get to Swedenborgianism's truly "eccentric" part. From Wiki (and yes, given the source, if it's wrong, please let me know; it's right as far as I understand):
At the age of fifty-six he entered into a spiritual phase in which he experienced dreams and visions. This culminated in a spiritual awakening, where he claimed he was appointed by the Lord to write a heavenly doctrine to reform Christianity. He claimed that the Lord had opened his eyes, so that from then on he could freely visit heaven and hell, and talk with angels, demons, and other spirits. For the remaining 28 years of his life, he wrote and published 18 theological works, of which the best known was Heaven and Hell (1758), and several unpublished theological works.
Why is this important? Because George Washington, father of our country, explicitly WELCOMED them to a place at the table of America's "implicit Protestant establishment," its "civil religion." As he wrote to them:
To the members of the New Church at Baltimore.
It has ever been my pride to mind the approbation of my fellow citizens by a faithful and honest discharge of the duties annexed to those Stations to which they have pledged to place me; and the dearest rewards of my Services have been those testimonies of esteem and confidence with which they have honored me. But to the manifest interpretation of an over-ruling Providence, and to the patriotic exertions of United America, are to be ascribed those events which have given us a respectable rank among the nations of the earth.
We have abundant reason to rejoice that in this land the light of truth and reason has triumphed over the power of bigotry and superstition and that every person may here worship God according to the dictates of his own heart. In this enlightened Age & in this Land of equal liberty it is our boast, that a man's religious tenets, will not forfeit his protection of the Laws, nor deprive him of the right of attaining & holding the highest offices that are known in the United States.
Your Prayers for my present and future felicity were received with gratitude; and I sincerely wish, Gentlemen, that you may in your social and individual capacities, taste those blessings which a gracious God bestows upon the Righteous.
John Adams likewise had a similarly lax understanding of America's "implied Protestant establishment." When explaining to Thomas Jefferson how American independence was achieved under the "general principles of Christianity," he then further clarified exactly what kind of "Christians" were united under these principles:
There were among them Roman Catholics, English Episcopalians, Scotch and American Presbyterians, Methodists, Moravians, Anabaptists, German Lutherans, German Calvinists, Universalists, Arians, Priestleyans, Socinians, Independents, Congregationalists, Horse Protestants, and House Protestants, Deists and Atheists, and Protestants “qui ne croyent rien.” [Protestants who believe in nothing.] Very few, however, of several of these species; nevertheless, all educated in the general principles of Christianity, and the general principles of English and American liberty.
Not just unitarians [Arians, Socinians, and "Priestleyans"] but universalists and deists, atheists and Protestants who believe in nothing have an equal place at the table of America's implict "Christian" establishment. Adams, who generally was an anti-Roman Catholic bigot, was generous enough to name them too as "fitting" within America's civil religion, its "IPE."
And even though Roman Catholics faced tremendous bigotry in America from the Founding era onward, presently (at least since JFK's election) they are accepted at the table of America's civil religion along with the Mormons, Jehovah's Witnesses, Elizabeth Clare Prophet, the Swedenborgs as just another eccentric Protestant sect.
We've truly come full circle. Welcome to "authentic American religion" 1776-2009.