Mike, your David U. McKenzie doesn't appear on the rolls of any Arkansas regiment, but you might want to check those Union regiments which operated in western Arkansas, especially Kansas and Missouri regiments, also (less likely) those Missouri Confederate regiments which served in Arkansas (they recruited heavily in northern Arkansas, but not so much south of the Arkansas river).

A fairly significant number of Confederate deserters and stay-at-homes joined Kansas regiments, especially in western Arkansas. An entire company in the 2nd Kansas Cavalry was composed of Arkansans.

You might also want to consider the possibility that he and his family became refugees and left the state until the fighting ended.

Good luck with your research.