Now that you put in the name Joshua Odell, I could look.
Just as the DAR entry notes, p. 25 of _NY in the Revolution as Colony and State_ does list a Joshua Odle as being among the ranks of Privates in the 1st NY Regiment, Continental.
However, I can not find a thing in the actual Regimental rosters, and the name is not indexed as being among those in any Continental regiment.
Since, as you pointed out, so much original material was destroyed in the fire at the NY State Library (after the books were published that extracted material there), it is a difficult or impossible task to reconstruct where that particular book entry came from. Other published material on the 1st NY does not list Joshua Odle / Odell.
It is possible he was mentioned on some document that no longer survives, but the Company muster rolls and pay rolls that survive are pretty complete except those for the first incarnation of the 1st NY Regiment before December, 1776. It was the Second Establishment from that time when Col. Gosen VanSchaick was in command of the regiment. So since you have no information on a date of service, there is no evidence for stating that Joshua Odell/Odle was in the 1st NY under Col. VanSchaick.
I am sorry I can not contribute to solving this puzzle.