Hi Candace,
I checked several sources and there does not seem to be a connection. John Swift Sr appears to have had only the one son John Jr , and no info on whom he married , if indeed he did marry.
I don't know if this helps: Alexander Swift 1480; Richard Swift 1503 m Jean Wheelock; 1530 William S. m Elizabeth Mason; 1561William S. m Jane Wilson; 1589 William Swift of Sandwich m Joan Sisson/Joane Basset; 1619 William Swift m Ruth Tobey. Thanks Candace