I don't have a basement or heated garage to store paint in during the cold Chicago winters so I end up bringing it inside the house for the winter. Since my chest freezer recently stopped working, I thought I could move it into the garage and store paint cans in it. Would this work to keep the paint from freezing or even getting too hot in the summer?

You know that very well may work as a freezer is insulated. The cold is the problem not the heat. I would give it a try and see what happens.