Welcome to the Forum. If it was dripping or slight stream (very small) at the faucet, then the water line shouldn't freeze. But if there is a leak or condensation outside the pipe or connection, that water will freeze on the outside. If there is a leak, fix it - If not a incandescence light bulb may help a lot in a small enclosed area.
Also, I always thought only the actual temperatures were involved in freezing pipes, not any wind chill value.
I live in SW Missouri. These temperatures (+5 - -10) seems to happen every couple of years. I am in Branson - all bedrock. Some of the water line can freeze. There is a place it is only 6" deep. My line seems to be ok down to about 5 degrees. I always trickle water at night when it goes below that. But if there is no leak and the enclosed well head, a good flood lamp should work at that point. You could even put a thermometer in there to see if it is warm enough. Best Wishes, Welcome to TBN