According to the vital records of Windsor, CT, Begat Eggleston was "nere 100 yer ould" when he died in 1674. This seems slightly unlikely to me, in part because his youngest child, Benjamin, was born only 21 years before. Then again, perhaps Begat lived up to his name and kept on begetting until he was nearly 80.
Begat fathered at least 7 children while living in Windsor: Thomas (b. 1638), Marcy (b. 1641), Sarah (b. 1643), Rebecca (b. 1644), Abigail (b. 1648), Joseph (b. 1651), and Benjamin (b. 1653). It is entirely possible that he had other children born when the people of Windsor were living in their first settlement (Dorchester, MA) or their place of origin (Dorchester, Dorset, England).
In Albion's Seed, David Hackett Fischer argued that Puritan parents occasionally named their children by opening the Bible at random and placing a finger at random on the page. I'm not sure whether I believe that, but it would be a good explanation for Begat Eggleston.
Sadly, while Begat had many heirs, he had no namesakes.