One thing that may help with a fridge that sometimes freezes things - put the sensitive things near the door. It's the back of the fridge that is coldest.
I refrigerate a lot of things that could be stored at room temp because I live in southern California, and unless I run the a/c all day when I'm not home (*not* going to happen), in the summer things are sitting out at over a 100 degrees, which is not what they call a "cool, dark place."

It's a short period of time when the temps in my apartment are "cool" during the day. Texas might be similar
I assume you mean winter squash? I wouldn't think refrigerating (keeping it away from freeze zones) would hurt it. And sometimes some batches from the store just mold more than others.