Just trying to bully you. If your position is strong and not wrong and have a strong verbal ability you can challenge them to counter sue. Coz they are shady and if you can provide a paper trail (that they sent you to work modifying your name and knowingly), they will pee their pants if you make a strong point even over the phone.
Your weakest position seems like : you are a student and want a stable status and job.

If you think you are ready and able to pick a verbal fight go for it, but prepare. If you think its not worth it move on. Record if they or their lawyer calls you. Build a position and argument and negotiate as your situation seems fit.