> Surely the FBI is not allowed to publicise random dirt they find on innocent people?

If they're doing an investigation, they very likely got the employer involved in order to get more information on the person they're investigating, and companies have liaisons for law enforcement, as well. If the FBI comes knocking and says, "we think you've hired a terrorist," it's going to ruffle some feathers at the company no matter how unfounded or untruthful the claim is.

It isn't just the suspicion of terrorism that might have law enforcement or the FBI knocking at an employer's door. If someone is suspected of any type of cyber crime, the FBI will be coming for all of their computers and electronic devices, including the ones they use at work.