William Eldridge died of debility at Camp Douglas, he is alternately noted as a member of the VA Home Guards and as a private citizen of TN in prison records.

John Eldridge of the VA Home Guards also died at Camp Douglas, his prison records are nearly identical to William Eldridge with the exception of his death date(Dec. 23, 1864) and cause of death(Dysentery).

Both of these men were originally buried in the Chicago City Cemetery before being removed to the Oak Woods Cemetery.

Sources:
N. A. Microfilm Publications M598 and M918

The death dates for William & John Eldridge of the VA Home Guards are the same as these two men listed on the Confederate Monument at Oak Woods Cemetery.

John Eldridge - 55 N. C. Reserves
Wm. Eldridge - 55 N. C. Reserves

http://graveyards.com/IL/Cook/oakwoods/confederate2.html