There are a couple of issues with your statement above, if I may; and perhaps you can better explain some of the thinking behind these assertions.
"Yes they were Puritans. But the people who put them togehter may not have been. ..."
This "sounds like" you are saying the "marriage of Ann Hackley & John Rice was arranged.
Please elaborate.
"My ancestor left John Rice in the Care of Puritans named ALLIN or Allen. "
Unproven thought, no known facts to support it. Surely the minister of Dedham knew his parishioners. That is ALL we know.
"And then the DNA trail goes south to Virginia, Barbados, and North Carolina."
DNA does not work backwards in time. If you have DNA matches to people who trace their first known ancestor to Virginia arrivers, it suggests there is a common ancestor, perhaps in England, and perhaps "long before" Puritanism developed.
"The Perrott Connection of Prunella & John the Quaker is still very much alive as the Fox family is also in my family of names."
I would be less skeptical if there was one date stamped and unaltered list of names instead of this continual addition to them.
Also, in building my family tree, I have found pursuing "names in the family" a singularly un useful way to work. Too many name changers.
Geography, occupation, social class, religion ... These have been far more useful attributes.