Based on the information and sources identified it is not likely that Rice was born in America. Between 1621 (post pilgrims) and Feb 1630 there were only a few ships that dropped of about 45 people, so this is also a factor in narrowing down where Rice was born.
Also, some sources use Rice and some use Richard which complicates the research. My ancestry subscription has expired and I will renew later in the fall and want to spend the summer doing independent research.
wish I contribute more.