Can i sue someone for giving false information and getting me fired.

If someone deliberately lied about you to another, you may have a claim against them for libel (written defamation) and/or slander (oral defamation).   This means an intentional misstatement of fact, not a statement of opinion.  Generally, the statute of limitations on a defamation claim is short, for example in New York it is 1 year from the defamation.  Also, if someone was negligent and made a mistake about you, you may have a claim for negligent misrepresentation depending on what that person's relationship was to you.  I don't know if you could successfully assert such a claim against a co-worker in MO.