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Taping the conversation in CA would not be prudent for the reasons outlined here: http://www.rcfp.org/taping/states/california.html
If you and your attorney do not agree on strategy, yes, the attorney can drop you as a client. There is nothing wrong or illegal about that.
Ultimately, it is up to you to decide whether or not to accept the settlement. It's important to understand the "why" part of the attorney's recommendation. Before making your decision on to settle or not, you really need to make sure that your decision is an informed one and that you understand the logistical risks of going to trial.
On a wrongful termination case, if the settlement is only $20K and assuming that the attorney will only get 1/3 and the attorney is suggesting that you take it, that suggests to me that there's a problem in the case and you are at risk of getting zero at trial. Really take the time to understand why your attorney believes that a problem has developed in the case (don't focus on the money, focus on understanding the problem in the case before you make your decision).
Good luck and best wishes. I hope that you find this information to be helpful and this answer to be ACCEPTable!