Considering that the sources of the "John (the Fugitive) Greene" legend (Cardinal John Morton/Sir Thomas More and Polydore Vergil) were all known to make stuff up out of thin air with the deliberate intent of discrediting the previous regime, there is reason to doubt that there ever was such a person.
If More did not actually work from a manuscript begun by Morton, he certainly listened attentively to everything Morton told him and remembered it well. (More started as a page in the Morton household.) *Morton was probably Richard III's bitterest and most dangerous enemy - and he wasn't going to let a little thing like Richard being safely dead stop him from continuing to blacken his character.*
Morton, as Chancellor of the Exchequer to Henry VII, was also responsible for a nasty little bit of chop-logic known as "Morton's Fork", which ran as follows: someone who was ostentatiously rich could certainly afford to pay the King's taxes; and someone who lived frugally must have enough saved up that they could also afford to pay the King's taxes. Heads the King's treasury wins; tails the taxpayer loses. https://en.wikipedia.org/wiki/Morton%27s_fork
More could also be a nasty customer if someone got on his wrong side - he thought "heretics" (Protestants) *deserved* to be hunted down and burned at the stake - and did at least some of the hunting down and sentencing himself. http://www.historytoday.com/john-guy/sir-thomas-more-and-heretics
As for Vergil, King Henry paid him well to write "history according to the Tudors".
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A further consideration is that if "John Greene, the Fugitive" was a real person and a Dorset person, he was not particularly likely to be connected to the Greenes of Isham (a Northamptonshire family).
There was an earlier - about a century earlier - Thomas Greene of Isham, who married Ala, daughter of Sir Anketil Mallory I of Kirkby Mallory and sister of Sir Anketil II. Her brother(? or father? it's often hard to tell which one is meant in various records) settled a manor of Sudborough on her and her husband circa 1360. Thomas and Ala had a son John, who was in possession circa 1400, married a woman named Isabel, and died before 1445. Sir Thomas Mallory of Papworth St. Agnes (Candidate #2 for authorship of the Morte d'Arthur), a grandson of Sir Anketil II, dispossessed Isabel and her son Thomas of the manor of Sudborough, but Thomas Greene later recovered possession (possibly after Thomas Mallory's death in 1469).
Obviously, *that* John Greene cannot have been the "John Greene, the Fugitive" of Tudor legend.
