Generally speaking, the fact that he was unlicensed/uninsured, (and same for you) does not change how the accident occurred. You have the right to sue anyone, but you would need to prove liability (how the accident happened) as well as damages (injuries, property damage), and causation (that the damages were caused by the other guy's negligence).
I would contact a local personal injury attorney and ask for a free initial consultation. They will probably help you figure out what you should do next. Good luck.
THIS RESPONSE DOES NOT IMPLY OR CREATE AN ATTORNEY-CLIENT RELATIONSHIP. NO ATTORNEY-CLIENT RELATIONSHIP IS IMPLIED OR CREATED BY RESPONDING OR FAILING TO RESPOND TO THIS RESPONSE. THIS RESPONSE DOES NOT PROVIDE LEGAL ADVICE. FOR LEGAL ADVICE, YOU SHOULD CONSULT AN ATTORNEY.
When you pay the ticket you received be sure to plead "nolo." you can do this at the office of the JP. This will mean that the ticket could not be used against you in a civil lawsuit should there be one.
Enterprise will also go after the person who actually rented the car because that person violated the rental agreement.
Unless the investigating officer saw the wreck the accident report in most instances cannot be used against you in a lawsuit. Enterprise will likely run a credit check on you and might file a lawsuit against you if they think you have assets to pay a judgment. Otherwise, they likely will not incur the cost of paying a lawyer to file a lawsuit. If Enterprise does, you need to talk to a local attorney to determine the cost of defending a lawsuit or the risk to your credit or assets if you do not defend the lawsuit and a judgment is taken against you. That is the risk you run when you do not have liability insurance.
Texas has comparative negligence. This means if two people are negligent the one less negligent might recover his damages reduced by his % of negligence. Enterprise knows this and will take this into consideration in deciding whether to file a lawsuit against you.
Even if Enterprise does not file a lawsuit be prepared for repeated letters asking you to pay.
An initial question is whether Enterprise is pursuing its renter as well as you. I would think so since the other driver would be in violation of his rental agreement for having an expired license. I have represented renter car companies against renters for violations of the agreement that resulted in damage. Enterprise could sue each of you for the damage and, if you did not settle, a jury would have to apportion the responsibility between the two of you. For example, if you were each found 50% at fault, each of you would be responsible for 50% of the damage. If he were found 80% at fault for speeding and you were 20% at fault for not yielding, you would be liable for 20% of the damage. This is a general rule only. There is a concept called "joint and several liability" where either of you could be responsible for the other's portion of the damages but that's beyond the scope of this response. I primarily defend cases and what I would argue, if I were you, is that Enterprise is equally, if not more, at fault for the accident by negligently allowing an unlicensed driver to rent a car. This is called "negligent entrustment." If you respond to their demand, I would drop this argument in your response. It may not scare them off but it could make them negotiate their demand. Most rental companies (and insurance companies) will work out a payment plan with you for a reduced amount if you decide to make an offer to pay some of the damage. For example, you might get them to agree to accept $3,500 paid out at $50 a month. While you will be paying for a long time, it does appear you bear some responsibility for the accident. I do recommend you contact an attorney and see if you can get representation at a reasonable rate. The attorney likely will require a retainer since you do not have insurance to pay his bill. You can also try the local bar association or law schools to see if there is any legal clinic or program that could assist you. I'm not aware of one but it's worth a try. Good luck.
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