Store in a cool, dry place. The best storage squash have a thick rind – butternut and blue hubbard usually store the longest. Honeynut, delicata, spaghetti, acorn etc. should be used earlier on. The adventure squash is in between. Any of the cut squash you received or cut yourself, should be cooked and enjoyed or cooked and frozen within a few days.
With all of these items check them frequently. Any of them that have a spot or blemish need to be used soon.