Puritan Massachusetts is often described as a theocracy, but it was really a direct democracy.
By now, in the 21st century, we equate democracy with liberalism; that is, we conflate direct democracy and liberal democracy. But they are very different: in a direct democracy, the people as individuals enter into a covenant to govern themselves directly, without representatives; in a liberal democracy, the people elect representatives to govern for them.
The U.S. today is a liberal democracy. But its roots are in the direct democracy of the Puritans.
Puritan religion did inform Puritan government, in that the direct democracy of churches was adopted for civil government. The congregation was the ruling body of the Puritan religion. They had rejected presbyterianism, in which a governing body makes policy for all churches and church attendees. Each congregation was its own discrete polity, and the people of each congregation handled all church business, from calling a minister to disciplining members to moving the church (as was common in early Massachusetts). As the towering Puritan minister Thomas Hooker put it, "Every man hath a right to meddle with the Congregation whereof he is a member."
And so every citizen had a right to meddle with the government of the colony. In the first two years of settlement, every adult male was made a freeman, and expected to attend General Court. As the population exploded, this was switched to a representative format, with freemen in each town electing two representatives. But non-reps showed up frequently at Court, and often protested laws made at the Court, and got their way. Each citizen felt a right to meddle.
This system frayed as the colony grew and the influx of non-Puritans ate away the sense of shared political purpose, and by the time a royal charter and government were imposed on Massachusetts in 1690, the people accepted purely representative government (liberal democracy).
But that urge to have each voice count was not easily abandoned. When the fledgling United States met in Congress to create a government in 1777, they created a direct democracy with the Articles of Confederation. As far as they could possibly do it, the Founders maintained direct democracy for the states, allowing their power to override that of the federal government. This system, despite its increasingly evident flaws, was maintained for 10 years until a sense of crisis led to the Constitutional Convention of 1787.
Even at this convention, anti-Federalists were loath to take power from the people in the states, and predicted that reps meeting in far-away Washington would lose touch with their constituents, banishing the last chance for direct democracy. The Federalists took this up as a good thing, stating that a more disinterested government at the top would encourage healthy faction at the state level, allowing for fresh influxes of new ideas from time to time as the people in the factions elected new reps.
But that desire for direct democracy that leads Americans today to call for less government and to covenant themselves with the NRA or AARP or other political organizations has its roots in the Puritan right of each individual to meddle.