Growing Vegetables in Upside Down Gardens?

I think it would, my father planted his tomatoes in 5 gallon buckets like the upside down tomatoes for years. He just cut a hole about 3" around in the bottom. Put the plant up through the hole, layed a piece of 1'x2' 6" long along both sides of the stem and tied them together securing the plant. The boards held the plant in the bucket when he turned it unside down after filling it half way with soil.

I fear your squash would be too heavy when it gets several squash on it but it might be worth a try.
I grow my squash in a big plastic tub. The kind nurseries get trees in. They have empty ones you can get for nothing usually. They have holes in the bottom for drainage. Good luck.