While the state of Connecticut may fine you and even imprison you for up to 90 days for not registering your car, the insurance company will not enforce that law for them. As long as you were insured at the time of the accident and you are not at fault, the other driver's insurance company should pay the claim.
You should, however, register your car immediately with the state. While the insurance company cannot refuse to pay based on the registration of your vehicle, they can report to the state that your vehicle was not registered. It will be much better for you if you remedy the situation yourself first.
The state of Connecticut is very strict about your car being registered in the state if it is your primary residence. This is due to the car tax that they press onto their residents. However, your vehicle registration has nothing to do with your insurance coverage.
If you have additional questions or the insurance company is refusing to pay, contact a lawyer for assistance. An attorney can ensure that you are paid either by the company or by the driver for your damages and medical bills.