I too have found the Jaycocks very frustrating, and everything you find out about them is generally suspect; especialli in the IGI; although there are some good things. I forget if I gave you the URL for, "Oak Leaves". At that site I have extracted the baptismal notations of Francis and Grace's children in Stratford-upon-Avon and nearby which should confirm William's date-of-birth approximation; a couple of mentions of Francis involving the church court. A hint of Francis Sr. in Stratford, CT; and the few mentions of William, Thomas and Francis that I could find; as well as a version of the 1698 census.
What you wrote sounds very probable; that William was with his sons in Hempstead. Do you know about their venture on the Delaware where Pittsburgh (I think it was Pittsburgh) is now? Of course, the focus of my research was to identify Catern (Katheryn) Jaycocks and her parents.
I also summarized the Jaycocks I found in the IGI in Warwickshire from about 1580 to 1630 in one of the Flewelling Forums. I'm sorry I can't remember which, and if I try to find it I may end up not being able to finish this message. If you need it and can't find it, let me know. While many of them were probably relatives of Francis Sr., I found none who might be his parents.
I wouldn't be too sure that you will find anything in Quaker records. It appears that Quakers weren't too keen on keeping records at first. Also, the Jaycocks were probably connected with the Presbyterian church at Hempstead. The only reason to move there, really. Also the Quakers didn't show up in New England until 1658 and were promptly told to get lost. One of my ancestors, the Scudders in Salem, was whipped and banished just for being nice to them.
I am still tracking down the evolution of Quakerism on Long Island, and I suspect that it didn't get a strong hold until about the 1670's to 1680's. The point is that, while your Benjamin may have grown to be a Quaker (which seems to be part of the reason for the move to Westchester, Ulster and Dutchess Counties from Long Island), he may not have been born a Quaker.
On the other hand, nobody seems to have really done a hard examination of Quaker records, so they are still worth examining. One advantage in later years is that when a Quaker moved to a new Weekly Meeting, they carried letters of introduction, which were often recorded at the new Meeting; so there was a record of where they came from. Somewaht of a rarity when some of these families really get hopping about.
I keep hoping a couple of good wills will surface clarifying some of the issues.
Much of my material comes from genealogists of 50-60 years ago, and echoes of their work, so I can't always verify what they have said. For what it's worth, Thomas Jaycocks' wife is said to have been Ruth. There was an instance when Thomas Jaycocks and Robert Ashman were indicated as brothers-in-law, and this is sometimes taken to mean that Thomas Jaycocks married Ruth Ashman. Everything I have seen indicates so far that Robert Ashman was young, single and alone whenever he came to New England, and that it is more likely that Robert married Thomas' sister. Robert's widow is variously called Catern, Caterin, etc., and I do know that Thomas had a sister, Katheryn bapt. 1613, who would have been the right age. Another question is whether Francis Sr. came with his children, or some of his children came together. For some time I leaned towards the letter concept, but now feel that Francis Sr. is the Francis in Stratford, CT; and that he later went to Hempstead to live with Thomas Armitage, his son-in-law. Gee! I hope I have that right, I'm pretty sure that there is one of those old references that Francis gave Thomas Armitage his property in return for support for the rest of his life.
It was about 1656-9 that Thomas Armitage (who had at least two grown children) took a new young wife (presumably one of the Jaycocks)and was arguing that he really didn't mean to leave all of his wordly goods to his son, Mannasseth, as now he would have another family to care for.
Well, I am probably confusing the heck out of you. The Jaycocks are always good for a headache or two.
Thomas A. Murray