Dear Wayne -
I believe the name of the ship was the 'Assurance'.
The name 'Hunnicutt' seems to be an American original. It isn't commonly found as such in the UK. From what I can tell, the relationship to William Huncote is somewhat speculative. I've also been told that Augustine Hunnicutt was a member of a party brought over by Daniel Gookin in 1642, but I've no proof of that either. It seems possible, since John Hunnicutt married Daniel's neice, Elizabeth Warren. I'd love to get proof of either theory.
Good hunting!
Laura