Clunky raises a good point. If you've signed a contract already you should resign and def not stay an employee on a medical cert or anything as it makes you look like you're doing the wrong thing. Resigning and taking a new job does reduce your chances of a successful constructive dismissal claim but it's not impossible and you'd still have the EEO route anyway. It def makes more sense to take the new job and you're also mitigating any economic loss which courts look on favourably.
Please do not make any remarks about your employer on FB! Especially naming them directly. People have been fired for those ad quite recently too so there is precedent. Normally the company would have to have a pretty clear social media policy for this to work though. The lines between private and work life are very blurred these days!!