The mystery has been solved - Andrew White of Sudbury, MA and Nova Scotia was definitely not from the Mayflower White family.
The final piece of evidence is "French and Indian War Muster Rolls", Massachusetts Archive Collection, Vol. 98:93 (A Return of Men Enlisted For His Majesty's Service For the Reduction of Canada - 1760) in which it lists the following:
Andrew White, 13 December 1760, age 27, b. Ireland, of Sudbury
He also listed himself as Irish on the 1770 Nova Scotia census.
Thanks to William Warner for finding the final piece of the puzzle!