Sounds like you need to harvest your squash now! If the withering vines are due to cold weather you must harvest butternut squash before the first frost to avoid rot. A few tips: Use a knife to cut the stem from the vine leaving a few inches of stem on the squash. This will help to prevent rot as pulling the squash from the vine can damage the stem and provide entry for bacteria. Store the squash indoors in a humid spot for at least a week if you suspect the squash is not yet ripe. This is called curing the fruit before it goes into cold storage or before you cook it. So yes, your squash will continue to ripen after it is picked but only if stored in a humid location. Perhaps outdoors on a covered patio where the skin stays dry and frost free but the air is still full of moisture.  It is possible your squash is already ripe and ready to eat. A ripe butternut squash will have a more dull rather than a shiny skin and the skin will have hardened to the point where you cannot use your fingernail to make a scratch into the surface. Harvesting at the peak of ripeness is an art that you will learn with experience but even squash harvested early due to a coming frost can be used for baking, soups and stews. Waiting too long can allow rot to set in and then the squash belongs in the compost pile. Depending on your climate and date of first frost, September through early winter is the time to harvest butternut squash. Keep growing, Marianne Binetti