The dogs regenerate, but the handlers do not. You don't lose handlers very often unless they are too close to enemy missle units, but if things get very confused or very close, you can lose the whole unit, handlers & dogs. That's only happened to me once.

It can be a bit confusing at first because the unit card will give the number of dogs, but the casualty report at the end gives the number of handlers. So you'll see your unit card number cut in half, but the report will say the unit had no casualties.