Squire may have been a nick name. Often the Justice of the Peace in the area was called "Squire". My own great grand father was.
There are several books written about the Lively Family. I'm a descendant of one too. There is the main one Livelys of America and two supplemental volumes of Corrections and Additions. You may want to locate and use it as a Guide, as I have found errors, which naturally happened for the time period it was written, early 1980s.