There are a few signs that a butternut squash is ready to be picked. It should be a dark tan in color all the way to the stem. Lighter greenish streaks near the stem can indicate it is not quite ripe. With winter squashes you can do the fingernail test. Can your fingernail easily pierce the skin? If so, it is not ready to be picked. Yes, looking for a stem that is starting to dry is also a sign that the squash is ripe. If the stem is brown the plant is sending less nutrients to the fruit. However, if the other indicators are there it can still be picked. If a frost is forecasted for your area you should pick the squash anyway. It may be able to finish ripening off the vine. To store squash long term, they should be cured in a warm, humid environment for a week or two before being stored at cooler temperatures.

Here is some more information on harvesting butternut squash.

https://hortnews.extension.iastate.edu/faq/when-should-i-harvest-butternut-squash

https://www.canr.msu.edu/news/when_to_harvest_your_garden

https://ag.umass.edu/vegetable/fact-sheets/pumpkin-winter-squash-harvest-storage