Cut the squash into cubes and place the cubes into a glass container that has an air-tight lid.
Pour about 1/4 inch of cold water onto the bottom of the container. This will help prevent the squash from drying out while stored. Try not to put too much water as it could make the squash become soggy.
Take 1-2 paper towels and place them under water to make them wet. Then, lay them on top of the cubes covering as much as you can.
Place the lid onto the container and store it in the fridge. They will be good for 5-7 days.
Notes
Uncut Squash Lasts Longer: If you don't have to use your butternut squash right away, and it's whole and uncut, you can store it for 1-3 months at room temperature away from sunlight.
Save the Seeds: When cutting the butternut squash, hold onto the seeds to use them for future recipes like roasted pumpkin seeds.