Storage of Butternut Squash
Two factors affect the storage of butternut and other winter squash, temperature and humidity. From  Bonnie Plants :


  You’ll see best storage results when you stash squash in a cool, dry spot. For most winter squash, store at 50º to 55º F with relative humidity of 60 to 70 percent.


Obviously, the temperature in our refrigerators is much lower than 50°F - 55°F. And humidity levels can be all over the place, higher or lower than recommended. 

To further answer why temperature makes so much difference: winter squash and pumpkins are still alive even after being harvested and continue to 'breathe'. From  NC State :


  Respiration rate of fruit is most effectively controlled by lowering the temperature. For each 10°C (18°F) reduction in fruit temperature, the respiration rate is reduced by approximately one-half. Chilling injury can occur, however, to some fruits at temperatures below 50°F.


Also:


  Whether in a home, garden or commercial planting, special care should be exercised to protect harvested fruit from excessively high (>95°F) and cold (&lt;50°F) temperatures, asphyxiation, and mechanical injuries such as scratches, cuts or bruises. Not only are mechanical injuries unsightly, they also provide an easy entrance for various rot-producing organisms.