What are my rights if my employer made a payroll mistake and never took out a 30 minute lunch break for the entire 3 years I've worked there?

Yes, you have to repay the money, assuming that you do legitimately owe that--that  is, that it is true that you should not have been paid for that lunch break, and were only paid (it not taken out) by accident. A mistake does not give you the right to the money, any more than the company would have the right to keep your money if they had accidentally underpaid you.

If you do not agree to the company's repayment scheme, they could sue you to recover the balance. If you don't have an employment contract protecting your employment, they could also terminate you.