Update:
According to what their site says, the 10th Illinois Cav. reached Arkansas in force in August of 1863 with an earlier detachment reaching Cane Hill in July of 1862.
So it is possible that the John Anderson was in the state earlier. I read every man's name in the whole 10th searching for injury and the place it happened. I found exactly what you did. He was the only John Anderson listed in the whole site.
Something of interest to us in central Arkansas is that
the 10th Illinois Cav. arrived in time to be part of the Battle of Reed's Bridge which they called Bayou Metre in late August of 63. The also lost several men in the battle to take Litte Rock.
While doing this search, I also compiled a list of 12 men who died at Brownsville. The site is now on Hwy 70 heading east to Carlisle.
So, I may know more about the 10th Illinois Cav. right now than about anyone except you.
Thanks.
Linda