This is a gray area under the federal overtime laws. Yes, the employer can require you to stay on the premises for lunch. However, you are supposed to be completely relieved of duties. The regulation states: "The employee is not relieved if he is required to perform any duties, whether active or inactive, while eating."
You need to discuss your specific facts and circumstances with a local employment lawyer right away. You may have additional rights under state and federal law.
I agree with the previous answer and think there is a very strong chance that this is a violation of the Fair Labor Standards Act. Of course, I am assuming from these facts that you are not being compensated for your lunch break. If you are being compensated for this time, then the employer is within its rights to require you to stay on premises.
You should consult a lawyer who handles overtime cases. I, along with many of the lawyers who handle these cases will give you a free consultation to go over the specific facts of your case and determine whether it has merit. If your case has merit, then most of us who practice in this area will not charge you any fee to pursue your case. We get compensated by seeking fees from your employer. I would be happy to talk with you further if you are interested. You can call me at 786-464-0841 or email me at firstname.lastname@example.org.
Mr. Kodsi is licensed to practice law in Florida, Georgia and North Carolina. His response to your question is not meant to constitute legal advice and does not create an attorney/client relationship. Rather, his response to your question is given for the purpose of providing general information about the issues addressed by your question. In many instances, questions such as yours do not include significant and important facts that, if known, could significantly change the reply and make it unsuitable. Mr. Kodsi strongly advises the questioner to confer with an attorney in your state for a more complete analysis of the issues addressed by your question.