Hey so per a phone call to the EP there are no state regulations regarding leaving shanties on the ice. As they said, it's just not common so there haven't been enough issues for it to be addressed at a state level. But there could be regulations for a specific body of water, and that would require a call to the town, but there shouldn't be on any "designated public" waters.
Regardless of lack of regulations there are certainly "best practices", and that's assuming you've already considered the big reasons people don't leave shanties out in general: theft, fire, vandalism, sinking, accidents with sleds and ATV's etc. Putting your info (name, address, email, phone number) etc visible on the shanty is a good idea. Also putting reflective material on all 4 sides and making sure it's not painted a light neutral color is probably smart too.
