I don't know about New Hampshire, but the general rule for the Maine Bureau of Parks and Land on the public reserved lands is that one can camp most anywhere. Restrictions apply only to particularly fragile areas. That applies from the New Hampshire border to the boundary of Grafton Notch State Park.
There are probably special rules for Speck Pond, where AMC employs a caretaker. But staying at Carlos Col, Full Goose and areas in between has always been both possible and free, when I've gone through.
Weary