Their story is told here:

Capture of the J.H. Miller
Account of The Wonderful Achievement of Three Boys
(Johnny Jones, Church Price, and Bennie Riggs)
On the Arkansas River
by B. F. Riggs, Louisville, Ky born April 2, 1849

Confederate Veteran, Vol. 15, pp. 73 -77

It appears they were assigned on special duty and belonged to no recognized unit.

Dr. (Captain) John J. Jones Commander of the expedition native of Georgia during the war

They were subsequently enrolled with General Fagin’s escort surrendering at Shreveport, La. June 7, 1865
Church Price died before war’s end