"Clayton died at Camp Chase, OH. I need help getting a headstone for Clayton if you can help in any way. I can get it ordered, I just need a place to send it and someone to install it. The Alabama SCV has been no help. " - Richard L. Smallwood-Roberts
Per the above quote, you initially indicated Clayton needed a headstone @ Camp Chase.
Since Greenberry was documented as a deserter during the war, I don't believe he is eligiible for a marker from the Veterans administration, unless he later joined the Union and served it honorably until his discharge. I see no evidence Greenberry served in the Union forces following his oath in 1864. As he was a deserter from the Confederacy, I can see why the SCV has no interest in the matter.
A marker for Greenberry denoting his Civil War service or otherwise, would have to be privately funded.