The Puritans had a profound influence on U. S. culture, including its desire for political independence.

Broadly speaking, there were and are three Christian church polities: episcopal, presbyterian, and congregationalist. Episcopal polity is hierarchical: a church is overseen by a priest, who is governed by a bishop, who is ruled by an archbishop. Authority is concentrated in one man and is conferred downward by appointment. Presbyterian polity is republican: local congregations elect elders who jointly form a session. Sessions call or appoint pastors. So authority is held by the session, but it is made up of elected representatives. Congregational polity is democratic: local congregations vote directly on issues effecting that particular church, including calling or dismissing a pastor. Authority rests ultimately with the people of the congregation.

Puritans explicitly rejected episcopal polity, and favored either presbyterian or congregationalist. The analogy between church polity and political government was not lost on the powers of the day. When James I assumed the throne, Puritans in England were hopeful that his Scottish Presbyterian background would incline him more favorably toward them. James, who loved theological disputation, convened Presbyterian and Anglican ministers to hear them debate how English religion should be organized at the Hampton Court Conference. At the conclusion of this, to the great disappointment of the Puritans, James astutely observed, "No bishop, no king."

So the Puritans are philosophically disposed toward representative self-government. On top of that, the conditions of their Massachusetts Bay Charter gave them broad leave to govern themselves, so they have both the motive and the opportunity for it.

It's also worth mentioning that the Puritans believed all children, regardless of gender or social rank, should be educated at least enough to read the Bible well. This led them to establish schools in the northeast, both primary (in fact they made a statute that any community of 100 families had to hire a schoolmaster) and secondary (they founded Harvard for the training of ministers). The priority given to literacy contributes to the rise of print culture in the northeast of the United States, which is enormously important in the movement for independence. The broadsheets become the battleground of ideas, which the literate public is invited to follow and join.

I'm sure there are other significant contributing factors. Look at Perry Miller's The New England Mind. It's older (1939), but a seminal text.