There is a connection between the Abigail Churchill that did the sampler and Inspector General Sylvestor Churchill of the US Army ( who Fort Churchill in Nevada was named after.) They are 4th. cousins once removed. The common ancestors would be John Churchill and Hannah Pontus of Plymouth, 6 generations back. I do not know how Sylvester Churchill is connected to the Jerome family.
Sir Winston Churchill the English prime minister, who's mother was Jennie Jerome, was actually a Spencer and not a Churchill. There is a very distant connection of Sir Winston to John Churchill of Plymouth that being 3rd. cousins 10 generations removed. Bill, W.G. Churchill, can give you a better handle or details on that connection than I, if you are interested.
The Abigail Prather Churchill, who married Meriweather Lewis Clark Sr., and the Abigail Russel Churchill,the sampler maker, may be connected but I have no information on that. Abigail Prather Churchill is connected to the Churchill Downs, Churchills and there may be an expert out there somewhere on that family to help you out.