Richard Rice III who married Hannah did not have a son named John. He left a will and all children were mentioned.
Also, is there any proof that Richard Rice I was the son or grandson of Dominick Rice and Alice Hussey? So far, based on a limited set of tests, it looks like descendants of Richard Rice d. 1692 in Northumberland Co, VA are in group 7A of the Rice yDNA project. But only one person in the group (kit 139670) claims he has traced his line back 16 generations to Edward Rice, b. ca. 1495, Co. Kerry, Ireland. The other kits belong to a scattering of Rice men that have not been definitively traced.