If you go down the site about halfway, you will find a search function. Type Ray, or J.C. Ray, or his whole name and you will find a few leads to follow. There was a J.C. Ray wounded at Wilson's Creek and another in the 37th Inf. Regiment (may be the same man). Not much information on either; but it's a start.
Happy Hunting!
btw. McNairy County, Tennessee is over by Shiloh battlefield.