Since Henry Adams (of "The Education of Henry Adams" fame) is a descendent of John and J. Q. Adams, is there any record of Henry using a CoA?  If so was it based on the arms of his ancestors?

I don’t know if he used them, but Henry Adams was the author of “The Seals and Book-Plates of the Adams Family, 1783-1905,” one of the main sources I used for the article on the arms of J. and J.Q. Adams.  Somewhat bizarrely, Henry denied that the arms on the bookplates and seals were actually arms—heaven knows what he thought they were—but then all the Adamses seem to have been a little reticent on the subject of heraldry, even while using it left and right.  Or actually, being Adamses, right and right.