they weren't being persecuted they were a break away faction of protestants. they called themselves puritans because they didn't agree with the way the church was being run. they thought it corrupt. so they came to America to form their own church the way they thought it should be run. as john Winthrop said. they were setting up a city upon a hill. their intent was to be a model for society. when new colonies were formed they wanted to be the ones they used as an example.
sorry just read the last part of the question. the puritans stayed in control of their colony (religion-wise) in Massachusetts until the first amendment was passed in 1791.
"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances."
and in a way they did achieve their dreams even today politicians are still using the phrase "City upon a hill"
