The same goes (presumably) for those with the “No morale check triggered on losing hit points” trait (Deathwish, I think).
What’s the worst thing that could happen? Someone could hit them and they’d lose hit points? But they don’t care about getting hit on the head by a berserk hedge knight with a massive hammer, so why care about being engaged in melee by a bandit thug with a wooden flail (or about fleeing allies, for that matter)? Obviously it makes sense that they could still be spooked by monstrous beasts or otherworldly entities, but logically you’d expect mushroom/Deathwish characters to be impervious to all morale effects related to regular danger.