I don’t know that either Samuel or Nathaniel are proven as children of Humphrey Basse and Mary Bouchier - but I don’t know if they have been disproven either. Supposedly they had 15 children.
The Great Migration Project Begins lists no origins for Samuel. The claim for Nathaniel looks more assertive.
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Bob Barris Notes:
There is a will for Humphrey that has not been transcribed and difficult to read and one for his wife Mary (Marie) that is transcribed and she names Nathaniel as her son.. There are several records for a Humphrey Basse of Essex at the National Archives UK Between 1600-1617. It looks like he was deceased by 1616. Chancery: Inquisitions Post Mortem: C 142/413/66 and Court of Wards & Liveries: WARD 7/57/141 both look very promising.
Some believe Benjamin and Mary are Samuel's parents, It is possible that Humphrey and Benjamin were brothers and Nathaniel and Samuel are cousins. One way or the other I believe they are somehow related.
Nathaniel's line is from the south around Jamestown and very well documented with plenty of DNA. So far I don't know anyone from the Samuel line with DNA.