Legal Question in Employment Law in California

It depends on the details of the situation. Most commonly the employer owes a duty to defend the employee sued individually, but not always. Sometimes the employer provides the defense because they see it as being beneficial to be jointly represented and the additional cost is nominal. In those unusual situations the employer might "pull the plug" if the employee quits. There is no way to really tell for sure what your rights and possible results would be without knowing the details, even though more than nine times out of ten the employer has a direct duty to defend that would survive the employee quitting.

If you are being sued because of conduct that might be outside of a normal work relationship such as, for instance, your alleged sexual harassment or assault/battery, I agree that the company might require you to find your own attorney. If it's for a wage and hour issue such as meal breaks or overtime, the employer should continue with your defense. In any case, you may leave the employer if you have another place to work. Make sure to let the attorney know where to reach you and that you'll continue to cooperate as best as you can.