Before storing winter squash for the season, you will need to put them through a process known as "curing", which can either be done on the ground in the garden (preferable), or in a warm dry place in the house.
"Curing" means giving the squash time to realize they are no longer
connected to their source of water and nourishment, which will put them into a
state of final ripening. Instead of continuing to grow more, they will start to convert their
starches to sugars, which will improve their flavor and shelf life. Leaving them out in the garden is preferable because the sun will warm them and accelerate the process.
After cutting the winter squashes from the vine (leaving a couple of inches of stem attached to the squash), just lay them on the ground in the garden or field and let them dry there for 5-7 days in the sun.
After your winter squashes have cured for 5-7 days, you can bring them indoors. The ideal conditions for storing winter squash are often found in a basement or in a non-freezing garage – someplace where it will stay
cool but not cold. A stable temperature of 50-55 degrees F is ideal.
Storing winter squash for maximum shelf life also includes good air circulation, so it's best not to let them touch each other. This also helps prevent any decay that may hit one from spreading to the others.
As with any crop in storage, check your winter squashes frequently for signs of degradation like soft spots or mold. Compost any of these that you find.
Some varieties of squash taste better with a little age on them, and others are better eaten fresh.
Acorn and Spaghetti Squash are best eaten from harvest through about 10 weeks after harvest
Delicata Squash are best eaten from harvest through about 12 weeks after harvest
Kabocha and Non-green Hubbard Squash are best eaten from harvest through about 16 weeks after harvest
Buttercup & Larger, Green or Gray Kabocha Squash are best eaten after the first 4-6 weeks of storage, up through about 5-6 months after harvest
Hubbard & Butternut Squash are best eaten after 1-2 months of storage, up through about 4-6 months after harvest