I was working for this company for a year and a half and they fired me saying it was because of my background but they did a full background check out and said it came back good. I didn't get into any trouble prior or after the application.
In Florida you are generally an at-will employee. This means that an employer can terminate you for virtually any reason other thean discrimination, retaliation for filing a worker compensation claim, retaliation for filing a Family & Medical Act Leave claim, and a few other reasons. Even if the company did a background check, if they want to terminate you they do not need cause. You would have no rights unless you were discriminated against or had a claim under one of the other acts or reasons available in Florida.
I guess it depends on what they mean by background. If they fired you because of your race, age, sex, disability, national origin, association with a person with a disability, whistleblowing, testimony under subpoena, worker's comp claim, or other protected status, you may have a remedy. Otherwise, in Florida they can fire you for any reason or no reason at all.
If it is because of an expunged crime, then I would want to know how they got the information. While it's not illegal for them to fire you for an expunged crime, it is illegal for them to obtain expunged records.