Under FLSA? No.
The FLSA does not require breaks or meal periods be given to workers. Some states may have requirements for breaks or meal periods. If you work in a state which does not require breaks or meal periods, these benefits are a matter of agreement between the employer and the employee (or the employee's representative).
Under your state law? Maybe. I've worked in states where it was required and boy oh boy was that a tough one to enforce. Many workers (myself included) would rather forgo an unpaid lunch break and leave an half hour early. Check it out.
What is definitely required is that they pay you for all time worked. If you are working while eating at your desk they must pay you. They must pay you for all overtime, whether authorized or not. However, they can also fire you for working unauthorized overtime (after they pay you for it).
However, it is certainly within your manager's discretion or company policy to require you to clock out for a 30 minute break. Just make sure--if you aren't being paid, don't work.