The Sayres are strongly connected to the Ogden family, who have their own prominent history and are ancestors of Phebe Mason, who married Aaron Price, whose family has a connection to Connecticut Farms and Springfield, NJ during the American Revolution (still tracking down details).
The chart below shows the ancestry of my great-great-grandmother, Martha Price Wilcox (who married William H.H Wilcox). Martha was the grandmother of MY grandmother, Janice Colborn Koehler.
The Sayre family seems to have come from Bedfordshire, England. The name (Sayre, Sayer, Seare, Sayer, Sayr and Seayre) occurs as far back as 13 16, when William Sayer was a landholder in Hinwick.
More can be learned about the Sayre family in several books including Sayre family; lineage of Thomas Sayre, a founder of Southampton by Theodore Melvin Banta.