Frozen onions in root cellar
Normally, when you freeze onions or winter squash, you'd first at least chop them. Makes them much more convenient to use.

When you thaw them out, they will not be firm anymore. The quicker you freeze something, the less damage it takes in the process—I suspect your freeze was, unfortunately, pretty slow. Each freeze does more damage. So, you don't want to thaw then re-freeze.

You can't thaw and store; the damage from freezing will lead to quick spoiling if you try.

I'd suggest you transfer them to a freezer, to make sure they stay frozen. That'll give you some time to figure out what to do with them. You may want to defrost an onion and a squash (just one) to see if they're still usable or not—if it turns out they're too mushy to be usable, then no point in spending time on the others. Defrosting in the fridge is probably going to be easiest on them. Put them in a bowl; they will probably leak juice.

If you decide to store them long-term in the freezer, you'll want to properly wrap them (or use zip bags).