Yes, what they can do is inform the PSB that you don't work any longer for them because your residence permit is tied to your current employer.

But what you can do is not show up at the PSB so they can't cancel the permit in your passport. Just look for different employment at a better school in another province and ask them to sponsor a new residence permit. You have to do a medical check again perhaps and some paperwork, but then you might be done, and nobody - even your previous employer - can kick you out of China!

Therefore, be careful and don't hand over your passport to your employer or to the PSB!