John Adams was US president 1797-1801. He lived 1735-1826. His father, John Adams, was alive 1691-1761. His father was Joseph Adams Jr. (1653/4-1736) and his father was, you guessed it, Joseph Adams (1625/26-1694), the son of Henry Adams (1582/83-1646) and Elizabeth Edith Squire (1587-1672)
Ruth's father, Peletiah, was alive 1646-1725. His father was Lieut. Thomas Adams (1612-1688), the son of Henry Adams and Elizabeth Edith Squire (same as above).
So it's John Adams(US Pres.) (1735-1826) --> John Adams (1691-1761) --> Joseph Adams Jr. (1653/4-1736) --> Joseph Adams (1625/26-1694) --> Henry Adams (1582/83-1646)
My line would be Ruth Adams (1673-1727/8) --> Peletiah Adams (1646-1725) --> Lieut. Thomas Adams (1612-1688) --> Henry Adams (1583-1646). [Note correction regarding Ruth's date of passing]
So ... Ruth would be a second cousin (once removed) from Pres. John Adams. It's clear they never met. Perhaps Ruth and Pres. John Adams's father knew each other, I don't know. It was a smaller world back then, but communication is not what it is now. (In some ways it was better!)
The source is "Henry Adams of Somersetshire, England and Braintree, Mass." by J. Garner Bartlett (1927), "Genealogical Register of the First Settlers of New England" by John Farmer (1998), and "The Great Migration Begins, Immigrants to New England 1620-1633" by Robert Charles Anderson (1995). For Ruth, the "Vital Records of Chelmsford, Massachusetts to the end of the year 1849" seems to be the main source as well as various NEHGS papers. They all cite Peletiah Adams and Ruth Parker as parents of Ruth, and Peletiah's parents are mentioned, too, with the birthdates squarely lining up with the other published genealogies of Henry Adams's descendants.
I have not seen these sources personally, but many people cite them in their genealogies. And most of these sources do not seem to be copying each other; rather, they appear to be independently verified.
As for Ruth's son, William Blanchard, people are citing New England Register vol 93, page 170. I have not seen this in person either. Questions remain, but I think this is basically true.
My interest started last year. I am new to all of this. Thus far I am a "gatherer" rather than a "hunter." I have been a fan of US presidents John Adams and John Quincy Adams, so it was a most pleasant surprise to find the family link. Perhaps in the future I will "hunt" down this info and have it properly sourced, so there may be less doubt. Personally, I have no doubt.
Please share any pertinent info about the Brown and Blanchard family ... I am interested.