I am a Pennsylvania lawyer, so I don't know the specific laws that might apply in NH, but it is an important question that applies to many other tenants.
When a landlord instructs the utility to turn off service, especially in the winter, that constitutes a constructive eviction. In Pennsylvania we have a statute that prohibits the utilities from turning off service without notifying the tenant and giving the tenant an opportunity to pay for the last 30 days' service. You may have such protection in your state. Consult a lawyer.
There may also be a state agency that protects consumers in utility matters. In PA it is the Public Utility Commission. If you experience an illegal shut off, contact them.
Clifford L Tuttle, Jr
Attorney at Law