Can I sue a minor/family for false sexual assault charges after the case was closed by police with no charges filed?

Perhaps you can sue, depending on the circumstances.  If you had not suffered so much harm from this incident, I would advise against considering a lawsuit.  However, you have been so seriously damaged that you may want to investigate a lawsuit.

Your claim would fall in what I call a "gray zone" of the law.  It borders on a few theories and each is highly dependent on how your state law defines the elements of the claim.  It is also very highly dependent on the facts and whether you can prove the facts (which, by the way, are not the same!  That you "know" something is not the same as you can prove it to a jury who know nothing about you, the child, the mother, the school system, the police, and everyone else.).

Before discussing the theories, it is worthwhile to assume that you will have to prove that the mother knew or at least should have known her child was lying.  Even if a certain legal theory does not require this, a jury will demand it.  A jury will be sympathetic to a mother defending her child against sexual assault.  If there is a question about whether you acted in this way, I promise a jury will resolve it against you.

Now that you know some of what you are up against as a practical matter, I can think of several legal theories you could consider.  They include intentional interference with contract (your employment contract), fraud, abuse of process (but, in most jurisdictions like Florida, this requires you have been arrested or your property seized), invasion of privacy, slander (which is verbally telling lies), and libel (which is putting lies in writing).  Your state may have a statute that prohibits false accusations like this.  If so, that statute may or may not be a basis for your lawsuit.  Those are the theories I can think of off the top of my head.  There may be others.

There is also a question of how your employer handled this.  You may have a claim for breach of contract (if your position was governed by employment rules and regulations that rise to the level of a contract), violation of civil service statutes and regulations (if they apply to your position), or some kind of discrimination on the basis of a protected class.  If your employer should have known the child was lying or failed to follow required procedures, you may have a claim.

I would encourage you to discuss this with your criminal lawyer to see if enough evidence exists to prove the child was lying, the mother knew it or should have known it, and/or the school system knew it or should have known it.  If so, this lawyer may be able to recommend a civil lawyer.  This is a tough case and far outside the garden variety personal injury case.  If you wish to investigate a claim, I would recommend starting with employment and civil rights lawyers.  They can give you some guidance and perhaps recommend a "personal injury" lawyer who is willing to step outside the typical automobile and property negligence arenas.

What a horrible situation.  I am sorry this happened to you.  You definitely have suffered damages as a result of this lie.  I would suggest you contact an attorney in your area who specializes in slander and defamation and seek a consultation.  There are many factors involved in your case and your options are likely dependent upon the state you live in.  A good place to start may be the attorney who represented you in the criminal matter.  He or she can probably give you some sound advice and point you in the right direction.  Good luck to you.