He very well might have been as there are no Finks in the listing of Union soldiers in Tennesseans in the Civil War.
If you ordered his service record file, see advert. above, it very might contain his home of residence.
The NPS S&S Sys. notes that this Richard J. Fink was a Farrier.
There were also two Robert J. Finks, One in Co. L, 3rd Missouri Cavalry and another in Co. G, 9th Missoui Cavalry.
George Martin