Hunting would have been impractical on a large scale. Hunting for game is much more complicated than you can imagine, especially for a moving Army and especially in the winter. I am not an expert in hunting by any means, but Arnold had several thousand men under his command in 1775 even though he lost several hundred during his failed attack on Quebec City on December 31st, 1775. My point is that he had a large number of people, and even if he had the ability to stop and have his men search for animals to kill, they would need thousands of animals to be killed to sustain them for even just a few weeks.

Armies need a ton of food and supplying them is very difficult. If you look at Washington's Army in the winter of 1775 - 1776 in Boston, his men were drinking 6,000 gallons of beer and 2,700 gallons of milk per day. They used almost 100,000 gallons of flour per month. (This is in an Army of about 20,000 soldiers )

The other things to consider around hunting is that it takes time to track game and skill to kill it. People also need to know these areas so that they don't get lost, and the vast majority of the Army didn't even come from these regions. The other fear would likely have been desertion. Even though many men's contracts were up at the end of 1775, letting men go off for hunting for a few hours would likely mean that some men would have fled, and they desperately could not have afforded that.