You can read the entire decision of Hale v. Everett here.
It's an interesting case; parts of it are extremely disturbing. It reminds me of Harry V. Jaffa's dictum that we have to view the American Founding thru the lens of its ideals, not compromises with those ideals or else we get a "pro-slavery" Founding and originalism loses its moral authority. Yes, that gets us "out" of slavery but also "in" to other places. Using Jaffa's method, this case gets thrown into the ashcan of history.
This case, decided in 1868, has to do with a Society of Unitarian Christians, formed years back. New Hampshire had a "Protestant Establishment" complete with a religious test that excluded non-Protestants from public office. A minority within the Church didn't like the new minister, who was more or less a deist. He claimed that he was a "Protestant" and defined said creed by anti-Catholicism. The NH Supreme Court agreed that anti-Roman-Catholicism was indeed one element "Protestantism," but something more. They also struggle over whether Unitarianism is a sect of "Protestant Christianity." Philip Hamburger notes something similar in Separation of Church & State that anti-Roman Catholic bigotry was one thing that united both the Calvinists and the Unitarians in the North East. Hamburger also shows that many anti-Roman Catholic bigots became champions of the principle of Separation, once Catholics started getting their hands on government aid. True, but there is also a long ugly history of anti-Roman Catholicism with INTEGRATION of Church and State in America, as this very opinion demonstrates. Jews, Roman Catholics and others couldn't serve as politicians in New Hampshire. That, it seems to me, is far worse than anything Everson ever brought us. That's why I argue the "religious equality" value needs to be the driver for the Establishment Clause. And even if the EC is unincorporated and neutered, let the Equal Protection Clause do the work. Religious equality means, at least, government non-discrimination for all religions and lack thereof.
James Madison was right when he argued government needs to stay as far away from involving itself in doctrinal issues.