John White - Sources say John was not a son of Gideon and Joanna

According to the Mayflower Society ("Mayflower Society of Descendants through Five Generations", Vol. 13, page 139), the source "The Big White Book" is in error, and this John's parents were not Gideon White and Joanna Howland. I've seen no original source, but it's mentioned in posts here:
https://www.wikitree.com/wiki/White-7127
and here:
https://www.familysearch.org/tree/person/collaborate/L1ZZ-MP7 (unfortunately the link in the latter post doesn't work)
- plus someone mentioned it in the About section on this profile.

If anyonce can find a source to establish whether or not they were his parents, that would be great.

As for Samuel White and Sarah (Brown) White, who are listed as his parents on WikiTree, I'm not sure they're the correct parents, either, since I've seen more than one source that listed their children without mentioning a son named John.