Kirby;

Lot's of Green's out there as you know.

Using the birds of a feather theory, several possibilities arise one of the better is Thomas Green, 2nd Lt Co E (Dunn's) 1 NE Cav. Appears he was recruited Schuyler County. I don't know enough of the activities of Dunn and company to figure out if this is a fit for time and place.

There is a Capt. Joseph E Green Captured Pulaski Co, Missouri Nov 20, 1862, died in captivity Dec 28, 1863. He is listed as with 3rd Missouri, no other unit info. He appears on a summary of officers in selected records.
https://www.ancestry.com/sharing/128653?token=yTzAqr%2BVeaQjSqSEQlP1S57%2Bta3Vmi3kWljjfrdyoxU%3D

Either of these ring a bell?
JJR