Haven't seen "John Dane's Narrative", but the extract given looks fake to me.
This all looks like the usual random assemblage of anybody in England with the same name.
Did the immigrant really marry a Greenleaf? It was a rare and localized name in England. But there seems to be some confusion with Greenfield (which suggests there's also confusion with the other immigrant).
There are baptism dates for two of the immigrant's kids. One is at Ipswich and the other at "Yarmouth, Suffolkshire", which presumably means Norfolk. But not the same place as Yarmouth IOW.
Are they real baptisms, and do they relate to known immigrants?
Are the immigrant's siblings known to be siblings?
Did the immigrant really emigrate in 1637? Nothing seems to be heard of him before the 1650s.