Why were Quakers banned from the Massachussetts Bay Colony in the mid-1600s?

Quakers were a persecuted group even back in the UK- they emphasized a more personal, direct relationship with God, unmediated by clergy, which was viewed as blasphemous and a threat to the established power of the Church of England. Because the monarch was the head of the church, denying one's allegiance to the church was akin to disloyalty to the state, and so 'nonconformists' like the Quakers were heavily proscribed.

Now, whereas 17th-century England was a state in which political and religious authority were heavily entwined, the Massachusetts Bay Colony was essentially a theocracy- although it was organized on republican lines, the franchise was limited to freemen, and one of the requirements for being a freeman was being a member of a Puritan church. This, too, was in the days when becoming a church member was a very serious business that involved an interrogation by the pastor to hunt down any hints of heterodoxy.

Because of the religious dominance of the government, the code of laws was heavily based off of Puritan beliefs. People could be prosecuted for crimes as various as playing dice or breaking the Sabbath (to say nothing of people executed for 'witchcraft'). Christmas and May Day were banned.

Back in England, the Quakers had been persecuted for their beliefs because blasphemy was seen as a threat to the social order. However, in the environment of Puritan New England, blasphemy was not just a religious injunction, but a civil offense. Simply by being within the colony and holding beliefs that were at odds with the established doctrine there, one was actively committing a crime, and it was on this basis that Quakers were banned.

They took no half measures, either. Any Quakers found on a ship coming into the colony were immediately imprisoned before being banished. There were a number of individuals who persisted in coming into the colony over a period of years in the middle of the 1600s, and four of them were eventually publicly hanged for their beliefs. This was seen as the last straw by the authorities in England, who revoked the ban on Quakers and, not long afterwards, ended the colony's de facto independence by sending over a royal governor to enforce the Crown's laws.