Yes, they can terminate you if they want to. Talk with your GM, be frank, explain what is going on. Try to work this out so that you keep your job. Tell them you want to keep your job. If they still go ahead and fire you, don't make an ass of yourself as you go out by saying you will sue them or something, because you can't. Thank them for the good time you have spent working for their company, assure them that this is a misunderstanding, perhaps ask if you can reapply when you have your record cleaned and this matter is resolved. In other words, be the professional, go out with dignity, and perhaps you'll be able to be rehired, or at least use them as a decent reference.
Then, as soon as you are terminated, you file for your unemployment benefits. If you have monetary eligibility, you have an excellent chance of being approved to draw. Because your wrongdoing was not related to your work, even if you did do it. Whether or not you did is not relevant. But if the employer wants to fire you for something that happened to you off the job, like being caught on one of those child molester predator shows, they can, but since it was not work related misconduct, you can likely be approved to draw unemployment as long as you are out and able and available to apply for other jobs.
If you stole checks from their business and passed them, then you don't get unemployment, as this is certainly work related misconduct.