If you are still interested in proving Squire was the one who enlisted in the 2nd NY Regiment, I might have some indirect proof.If you check with Fold3.com and look at Nicholas Sharts application, you will see that he mentions Squire Pixley and William Pixley being drafted in 1776. It is hard to read all the words, but the gist of it is that Nicholas names several men who were in that group which was drafted to help and they were all deceased by 1822.
Also Squire Pixley b. 1763 deposes in 1845 in Rhoda Pixley's widow's application.It clearly states there they this Squire was married ca. Sept 1783 and William and Rhoda were married Oct., 1784.
Since Squire Pixley was drafted in 1776, it would be unlikely that he was enlist in 1783.