Partial answer (other parts of an answer would be jurisdiction specific):
Would this constitute a slander lawsuit against the group of employees
and outside agency?
The employee falsely alleged to be mentally ill would be the plaintiff who would be the only person who would have standing to bring suit.
The person allegedly making a false statement to someone other than the employee alleged to be mentally ill (an action which is called "publication" of a false statement) would be the defendant(s).
Listening to an allegedly false statement about someone is not slander, nor is believing it to be true.
In the U.S., in most cases, knowledge of the truth or reckless disregard for the truth (or at least negligence in getting the facts wrong in private persons cases not involving a public interest) is necessary to establish liability. Someone who doesn't know that a statement is false and has some basis to believe it is true, can't be guilty of slander.
In many non-U.S. jurisdictions, slander liability is possible without knowledge that the statement made is false (and without reckless disregard for the truth or falsity of the statement).
Also, it matter who the "outside agency" is in this case. Governmental agencies are typically immune from liability for slander for statements made in the course of their official duties. Without a fuller understanding of who is saying something and why, it is hard to evaluate the claim appropriately.