There are a ton of campgrounds in the White Mountains in New Hampshire. There is free dispersed camping in the White Mountain National Forest and also camp sites to rent. It is a well traveled area. There are plenty of places that are off the beaten path but still close enough to civilization for emergencies. October is borderline out of season for camping though, so you will probably miss a lot of the crowds.

Just to reiterate what /u/hstarnaud said, Northern New England gets pretty cold in October, particularly up in the mountains. After a quick google search, the average low temperature for Franconia, NH on October 31 is 29F. You will want to make sure you have a good sleeping bag (Probably rated to about 0F) and a good pad to sleep on. I don't mean to scare you off, I have been camping in the dead of winter many times in the White Mountains, October is completely doable just be prepared for cold.

If you want to get off the beaten path in Massachusetts, you should stay in the western and north-central portions of the state. Anything within 2 hours of Boston is pretty much suburbs. You can pretty much draw a straight line of pure suburbia between Boston and NYC. It is part of the Northeast Megalopolis which runs from Boston to NY to Philly to Baltimore to DC. It is quite literally an unbroken line of pure suburbs, and most heavily urbanized region of the United States.

Come to think of it, if you are just looking for camping you might as well avoid Massachusetts and the rest of Southern New England all together.

Maine is awesome, you will have a great time at Acadia. I don't really know anything about Vermont, I hear they have cheese.