The DAR Patriot Index has his date of birth as 4 Feb 1746.
He may be the Josiah Remington in the 1790 census at Westfield, Hampshire Co., MA. David Remington is listed with him -- possibly a nephew, son of Daniel II. Also in the same town is Jabez Remington, possibly a brother or cousin.
The 1810 Federal Census in NY lists Josiah & Sinecer Rimmington at Cambridge, Washington Co. Seneca is possibly the son of Daniel II (not confirmed), and if true, would be Josiah's nephew.
The 1810 Federal Census for Cambridge, Washington Co., NY, has three Remington families or heads of families: Jabez, Josiah, & Severa (Seneca). Josiah and Jabez are listed together. Perhaps that is this particular Josiah (age 45+ in the census), and Jabez is Jabal, his son (age 26-44). These family groups need more research.