The New Hampshire Insurance Department states that New Hampshire law does not require you to have automobile insurance. However, if you have an "at fault" accident without having insurance coverage, you will be required to post a bond or cash equal to the amount of damage you caused in that accident.
If at fault in an accident in this state you must also satisfy the NH Financial Responsibility requirements to continue to operate a vehicle in New Hampshire. These minimum limits are 25/50/25 (Bodily Injury $25,000 per person; $50,000, 2 or more persons; and Property Damage $25,000). A $75,000 single limit policy will also satisfy the minimum requirement.
Even though your home state may not require auto insurance coverages typically other states do require you to have auto insurance to operate a vehicle on their roadways. Most states require either insurance or other financial responsibility be placed on your car that would cover an accident you may cause out of state (conform to the state laws of where you are driving) and this would be good enough to show as insurance coverage to not be cited for driving without insurance.
Massachusetts law requires that all cars have insurance on them with the minimum limits of $20,000/$40,000 for bodily injury and $5000 for property damage.
Compulsory motor vehicle liability insurance is discussed in Chapter 90: Section 34J of the MA legislature laws. Here it states that being without insurance on a motor vehicle is punished by a fine not less than $500 nor more than $5000 or by imprisonment for not more than 1 year or both the fine and jail time. There has been some added advice in the comments:
If the vehicle is driven in MA for more than 30 days in a year, or the owner lives or works in MA, he or she has to have insurance.
This Massachusetts law says that whoever operates or permits to be operated or permits to remain on a public or private way in MA a motor vehicle s subject to the provisions of section one A [section of MA law that defines compulsory motor vehicle liability insurance]. If at such time one is on the roadways of Massachusetts there is a need to have a motor vehicle liability policy or bond or deposit or the penalties we noted above would apply.
Massachusetts General Law (MGL) Chapter 90, Section 3 discusses non-residents and their need for liability insurance. When we spoke to The Division of Insurance for Massachusetts, they assured us that even as a resident of a state such as NH that does not require insurance you will need to have auto insurance in place to drive in MA.
Normally in MA a person cited for driving without insurance may also have their driver's license suspended for 60 days by the Registry of Motor Vehicles and likely their registration and tags to the vehicle suspended as well. Since you are a non-resident we would advise you to check with the MA RMV to see if these penalties apply to you or what penalties you would receive if found to be driving in MA without insurance though your home state does not require auto insurance be purchased.
Vermont is another neighboring state of New Hampshire. In the Vermont Statutes (Title 23, Chapter 11, Subsection 800) it states that no owner or operator of a motor vehicle required to be licensed shall operate or permit the operation of the vehicle upon the highways of the state without having in effect an automobile liability policy or bond in the amounts of at least $25,000.00 for one person and $50,000.00 for two or more persons killed or injured and $10,000.00 for damages to property in any one accident.
Subsection 801 of Vermont laws discusses when proof of financial responsibility is required. Subsection 3 in part states that penalties will not apply if the operator was a nonresident, holding a valid license issued by the state of his or her residence, at the time of the accident, satisfactory proof, in the form of a certificate issued by an insurance company authorized to transact business in the state of his or her residence, when accompanied by a power of attorney authorizing the commissioner to accept service on its behalf of notice or process in any action upon the policy arising out of the accident, certifying that insurance covering the legal liability of the operator to satisfy any claim or claims for damage to person or property, in an amount equal to the amounts required under VT law with respect to proof of financial responsibility, was in effect at the time of the accident. Just to give you an idea of another states law as well.
If you travel out of your state of New Hampshire by operating a motor vehicle you will in most all states need to have at least basic liability insurance on the vehicle that would cover damages you may cause to others. Most auto insurance policies include a broadening clause that raises the liability limits to the minimum of the particular state that a driver is operating a vehicle in and thus conform to the minimum limits of that state compare to their home state.
To better protect yourself, your car and your assets (such as your home, bank account, etc) we would advise you to carry auto insurance at all times, not just if you plan trips out of your state. If you are found at fault in a severe accident and are sued by the other party your personal assets would be at risk.
Before driving out of state without insurance on your motor vehicle it would be advisable to find out more about state insurance laws and if you need auto insurance (and if so what type) as a non-resident to operate a vehicle on roadways in MA, VT or any other state. You can contact the state's DMV or insurance regulatory body for this type of information.
Follow this link to get free auto insurance quotes for New Hampshire.