I have not found Greene's Company, this piece may be of some help.

During the Civil War Missouri voted to stay in the Union and those who fought for the South were treated as guerrillas and not as members of the Confederate Army. For that reason, Missourians who fought for the Southern cause were not accorded any military rank when captured. On September 30, 1864, Confederate Major General Sterling W. Price entered Franklin County Missouri and conscripted every man and youth found at home who was able to bear arms. Some 200 Missourians from Price's raid were captured by Union troops and sent to Rock Island. Leonard was one of these conscripts. He was captured on Oct. 29 during a skirmish near Mount Vernon, MO.
http://www.findagrave.com/cgi-bin/fg.cgi?page=dfl&GRid=9114808

.............

Pvt John W. Yates

Birth: unknown
Death: Jan. 9, 1865

YATES, JNO. W.
CONSCRIPT, MO
1/9/65

Burial:
Rock Island Confederate Cemetery
Rock Island
Rock Island County
Illinois, USA
Plot: Grave # 1760
Note, this is a Confederate tombstone