It doesn't seem to work this way (at least in the current 1.5 beta). Looks like normal damage and stun damage use separate health bars.
However stun batons are still very effective in close combat (at least compared to ballistic and laser short-ranged weapons).
By the way, congratulations on a great X-Com remake, Chris!
It is a very good game compared to original X-Com 1, 2, 3, UFO Afterlight, Aftershock and Aftemath, JA 2 and Firaxis XCOM. You did exceptionally well for your first project and I have a lot of fun going through my first Veteran run =)