What are my rights if I was denied a job after I was told to leave my other job?

You may have a claim under the theory of "promissory estoppel" (which is also sometime known as "detrimental reliance"). When--

1) Someone promises you something, like a job;

2) To accept the rpromise, you would have to do something to your detriment, such as giving up an existing job;

3) It is reasonable for you to rely on that promise; that is, it was plausible;

4) The party making the promise knew or reasonably should have known that you would have to do something to your detriment, such as leaving existing employment which they were aware of (i.e. they knew you had a job);

5) Knowing you'd have to do something to your detriment, they made the promise anyway; and

6) Acting in reasonable reliance on the promise, you did in fact take the detrimental step (in this case, leaving your job)

--then the promise may be found to be binding due to your detrimental reliance. (The promisor is "estopped" from not honoring his/her/its promise.) From what you write, this might be a case where you can enforce their promise (get a job; or at least get compensation) due to your detrimental reliance. It would be worth your while to consult in detail with an attorney about the situation. Good luck.