They cannot evict you by telling you to leave. They have to file an action in court after posting a 3-day notice on your door. Before they do that, send them a certified letter of non-compliance with the Landlord/Tenant Act, Fla. Stat. Ch. 83 and advise them they have seven days to cure the problem or you will withhold rent for the non-compliance.
Have you requested the pest control service in writing? Most apartment complexes require written requests.
If you can afford to move, I would do that. It's not worth the health risk, even if they resolve the issue in seven days.
This communication is not intended to create an attorney/client relationship. It is always recommended you consult an attorney in person to discuss your case. The Law Offices of Stage & Associates practices state-wide and represents homeowners and community associations. Please visit our website at www.stagelaw.com.