I recently discovered that Capt. John Gillespie Birney (1822-1870) is a 4th cousin and his final resting place has long since been re-discovered since this article was written in 2003. I decided to leave this information here just in case someone else researching the Birney family should come across this article.

The following two links provide additional photographs and information on the location of his grave site, along with his wife and son. Unfortunately, the grave site was never marked.
However, since I am a relative and Gillespie was a veteran, I believe I can request a U.S. government supplied headstone for him and his family. I will request a headstone and update this article.

Find A Grave memorial provides the GPS location
https://www.findagrave.com/memorial/66530190/james-gillespie-birney

Green-Wood Cemetery site provides the unmarked location of James G. Birney family burial location
https://www.green-wood.com/burial_results/index.php