The J.A. Simpson in the Missouri SOS listing in Co I 16th MO (Parson's Reg't) is listed as a resident of McAlester OK and came from an 1882 reunion roll. Not a fit for either John Anderson or James.
The J.A Simpson in the 2nd Missouri Volunteer Infantry (Capt Livingston) was 18 and listed as a resident of Mt Pleasant MO, Native of Kentucky. This listing does not fit either James or John.
The biographical profile of James Anderson is accurate but still does not give us the detail of John Anderson Simpson's Civil War service. Given his reported move time to Randolph County Arkansas it is a strong probability he served in an Arkansas unit.
Fristoe's Cav is a good fit but would like a bit better documentation or corroboration, thanks for contributing.
John R.