Yes, if you brought your mirror indoors after observing in sub-zero temperatures and dropped it into boiling water, it might break. So don't do that! In normal use, there's no chance whatsoever of damaging the mirror.
If you're going to spend a night car camping in sub-freezing weather, load liquids like beer, soda, and eggs into your cooler (a YETI works great). It keeps water from freezing or becoming ice cold on a snowshoe trip or hike. Plus, a great trick is to fill the bottle with hot water before bed.