[AtlUmpSteve]

You may consider this just semantics, but I believe rule terminology is important.

No batter is ever out for entering the dugout. If she/he is still a batter, you put her/him back up to bat.

However, after strike 3 is dropped, the batter becomes a batter-runner. A batter-runner is out when they abandon the effort to first base by entering dead ball territory (be it dugout, or go out another gate).