Hi Theresa, 

Under the very best of circumstances, butternut squash will keep for about four months in storage, but it's not uncommon to see them starting to rot as soon as two months after harvest. Unfortunately that's just the typical shelf life we can expect in home storage conditions.

From your description, it sounds like you've done a great job at providing the right storage conditions. After harvest, the best practice is to "cure" the squash in a warm, well ventilated space (about 80 F) for 10 days, then move them into cool storage at 55 F with 75% humidity. Keeping the squash dry and handling them gently to avoid bruising is essential.

You mentioned that you also washed the squash in a dilute bleach solution. The research on this point is conflicted, and some researchers have suggested it's better to wipe the squash down with 1:10 bleach/water solution rather than washing by submerging. Either way, it's important to dry the squash immediately after washing or wiping down with the bleach solution. 

Cardboard bins are frequently used for storing squash, so I don't think your choice of cardboard liners on your wire racks was a factor in your squash starting to rot, unless the squash were wet when you placed them on the cardboard and moisture became trapped. 

You also ask about the potatoes you have stored with the squash. Potatoes do not release much ethylene gas, so storing these two crops together is not a problem. Apples and pears do release ethylene gas, so you should avoid storing these with your squash.

You have taken the essential steps to extend the storage time of your squash, but unfortunately it sounds like they are reaching the end of their useable life. As soon as you notice softening or mold growth, get rid of those squash to prevent spores from spreading to the rest of the harvest.

Penn State Extension has a great summary of squash storage information if you care to read more: https://extension.psu.edu/prevent-rot-of-winter-squash-in-storage

Thanks for using Ask Extension!

Jared