My information came from my second cousin, Lloyd Dewitt Bockstruck. He has been doing genealogical work for decades. He would be able to tell you exactly what documentation he has. Type his name into a search and it should come up with a location to contact him or write to him c/o the Dallas Library. He is in charge of the genealogy division there. He lists John's father and mother as Richard and Agnes (Wingod) Briggs. He also lists Agnes as daughter of Richard Thayer, son of John Thayer.
Lloyd lives in Dallas, not very far from where John Kennedy was shot. I haven't been in touch with him for a while except for exchanging cards at Christmas time. He is always on the go, giving lectures across the country on genealogy, so it may take a while for him to get back to you.