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).

The first point: You cannot defrost and store them open again. The problem is that freezing destroys the cell walls and when they thaw, they will be wide open for all kinds of bacteria and mold to colonize them. There is no chance to keep them in the root cellar again.

You could certainly defreeze and cook them right away, but if you have lots of them, you will have to use a canning-suitable recipe and go through a complete canning process. Your post sounds like you are not inclined to do it now, and you can always do it later if you freeze them now. Besides, after the canning they will not be usable in the same way as fresh or frozen onions would. Any cannable recipes will add lots of acid, totally changing the taste and texture of the onions.

You can thaw them, slightly fry the puree, then freeze in cube trays. (Or puree from frozen if you have a high-powered blender). Store the cubes in bags. They work great in soups and purees, but less so in e.g. roasts.

Putting them whole into the freezer is safe. But when you take out whole bulbs for cooking, you will have to wait for the onion to thaw enough to be able to cut it, so it won't be convenient at all. So I won't go that way.

Thawing them now, cutting, then freezing, will work. But the thaw-refreeze cycle will break up their structure, so the thawed onions will be like mush. All the single pieces will also stick together during refreeze, giving you a big lump of onion. These two points make me think that pureeing is the better treatment.