He was elected MP for the seat of St Albans in England, in 1640. In fact he was elected twice in the same year, firstly for the Short Parliament the a few months later for the Long Parliament. This is all on record, I can't see how the same man could have died five years earlier in America.
See: http://books.google.co.uk/books?id=5V09AAAAYAAJ&pg=RA2-PA229&am...
Look down the list for the county of Hertford - St. Albans - John Jennings Knight of the Bath .
Please see:
http://books.google.co.uk/books?id=cnwiNctJrGgC&pg=PA450&dq...
It describes Richard Jennings, father of Sarah 1st Duchess of Marlborough, succeeding his father Sir John Jennings of Sandridge as MP. It also mentions Richard's mother as Alice Spencer.
Just to be entirely clear. The records, not inconsequential ones as they are records of elections and members of the English parliament, show that the John Jennings who died in North America in 1635, could not possibly be the same man as Sir John Jennings of Sandridge, who was elected to parliament in 1640.
Sir John Jennings was the husand of Alice Spencer daughter of Sir Richard Spencer of Offley and was the grandfather of Sarah Jennings wife of John Churchill 1st Duke of Marlborough.