can i sue a company for lying to avoid paying unemployment

i honestly don't know, but i would think so.

You should appeal the decision within the time frame allowed, and show unemployment the documentation proving that they lied. As to suing them for, say, defamation, under Florida law statements made to unemployment are considered within what's called a "privilege" and the courts are unlikely to allow you to bring such a claim against the employer.

I hope this helps!