It only happens during the night and only seems to target the growing shoots of my squash and melons. I went out at 10pm last night and just missed whatever did this, the vine was still seeping.

Not sure where you are...you haven't clued us into if you are city or country living. In summer small animals start searching for water tho they are afraid of open water, so they supplement with tender growing vines. Some of us leave water out with rocks or floating wood for small animals.
Me, I have deer that love to munch their thirst away, but I also have squirrels, coons, rabbits, and various other varmints that could nibble.