Perhaps the Amazeen family has Moorish roots in the Iberian peninsula.
When thinking about this theory, wonder what happened to the old Moorish families, who'd been living in Spain for centuries, during and after the Reconquista -- cultural/religious conversion, intermarriage, etc)?Could John Amazeen's great-great-grandparents (100-150 years prior to his birth; late 1400's to early 1500's) have these roots?
Could members of the family (John'sfather, grandfather, or great-grandfather) have started to enter seafaring activities during this period? Maybe John was a crew member of a cod fishing fleet and decided there were better opportunities in the new English colony. Amazeens could have migrated to England from Spain/Portugal during this period, and then to New Hampshire.
Perhaps sometime during the 8th to 15th century (the period of Moorish presence) Amazeen family ancestors migrated into Iberia/Spain across the Gibralter straits from Morocco; and even earlier over a longer timeframe from the Arabian peninsula-- originating in an area known as wadi Amazeen, which is near Aden in Yemen, across north Africa to Morocco.
Just speculation (a lot of "perhaps"in the above theory)