Can a company fire you for hearsay? And can a client bring up charges on you if they claim injury?

Unless you have an employment agreement, or sometimes an employee handbook, which specifies the reasons why  you can be fired, or there is some statute which provides that you cannot be fired for a specific reason (such as race, religion, age, etc.), you can be fired for any reason at any time.  Your client can decide to sue you, because anyone can sue anyone at any time, but that doesn't mean she will win.  She can also try to have criminal charges brought against you, but this is less likely.  As for whether you should sign the resignation letter, I think you need to consult with a lawyer in person, and give him or her more information than you can provide in this limited forum.  For what it's worth, and from the very limited information you have provided, it doesn't sound to me as if you should sign it.  It could not only be taken as an admission (which I would not advise you to sign even if you had done what the lady accused you of doing), but might affect your right to unemployment insurance and other benefits.