Unlike most garden vegetables where the fresher the better is the rule, butternut squash, like some other winter squash, actually improves with storage. See this NPR article for more information: Don't Squander That Squash: Like Fine Wine it may Improve with Age.

The squash continues to develop after harvest, changing starches into sugars and increasing the carotenoids. They'll be at their best after a couple of months.

Harvest butternut squash when they are mature by cutting the stem about 2 inches from the fruit. Cure them to harden the skin at 80-85 degrees in the garden for 5-7 days. Store them in a dry dark space at about 50-55 degrees.