I can't say for sure but your family could very well be a branch of my Green family. We had at least three branches move to Florida. We are still trying to figure out who was a son of who, and who was brothers/cousins ect. But it is obvious they are the same family.
The name Lewis and Elisha both appears in my Green family. Do you have a male descendant who can be DNA tested? Our Green DNA is very unique. We are the only Greens that have out DNA because as it turns out...we are not really Greens.
Personally, I would look into the family of Elisha Green. Elisha appears in the 1820 Bulloch County Georgia Census. In 1840, he is living in Ware County with a Daniel Green and Lewis Green. Lewis Green ended up in Hamilton County Florida, he did have a son named John but I do not have any information on his son John.
Elisha had a LOT of Children. He ended up in Baker County Florida where died but also lived in Columbia County. Elisha also had a son named John but this is well before your John Green's time. Elisha was married twice and had 19 children between them. Out of the 19, 10 were sons. I would start looking with him and see if you can't work backwards.
I am descended from these bunch of Greeens, but mine stayed in Georgia. I have been working with a descendant of Lewis Green of Hamilton Co. Florida. Both of our DNA's match.