Does a co-signer have the right to take away the car or ask for you to refinance only in your name because you broke up?

Yes, the cosigner is partly responsible for the car. Therefor they can take equal ownership or ask that you refinance in your own name so they are no longer liable if you default on a payment. Remember, missing payments will hurt both of your credit histories so they have a right to protect their reputation. * A cosigner has no legal right to any property unless his or her name is on the title, nor can a cosigner force someone to refinance a loan without taking the person to court and receiving a court order directing the primary borrower to take such action.