It's inconceivable that a man of military age in Baxter County would not have served in the Civil War. That was one conflict that didn't permit fence-riding. He had to have served on one side or the other. Since he doesn't show up in any Arkansas regiment, you might want to explore the possibility that he served in a Missouri regiment -- either Confederate or Union. A huge number of men from northern Arkansas served in Missouri units. In some regiments there were entire companies of Confederate (and in a few cases, Union) volunteers from Arkansas.

There were probably a lot of men named John Rose who served on either side in Missouri units, so you should start your search in those units that recruited and/or campaigned in northern Arkansas.

Good luck with your research. If you get lucky, or if you need any help, let us know.