I am adding a room onto my camp in Northern New Hampshire. We close the camp down for the winter with no heat at all, so the temperature in the camp gets below freezing. We have been told not to use sheetrock for the ceilings because of humidity and freezing which will damage the sheetrock.

Is there any alternative ceiling that will hold up to the conditions of humidity and freezing that complete closure of the camp causes?

Hardibacker board will hold up quite well in these conditions. You could also buy Hardiplank siding and use that along with painting. James Hardie: World Leader in Fiber Cement Siding and Backerboard

We close our cottage up every winter & never have problems due to cold. We've been doing that for about 25 years and have never had to patch or repair cracks in the drywall.

Moisture/ Humidity is more of a problem - not for drywall as much as mildew on carpets & cushions. Ensure the indoors gets lots of natural light. We recently started putting in chemical dryers in a couple of rooms (these are typically used in boats - check your local marina)

The biggest problem is mice. Make sure you house is absolutely 'hole & crack free' - particularly holes where wiring and plumbing enters.