I am a law school graduate currently awaiting Bar results. What I offer to you is mere information, not to be construed as forming an attorney-client relationship.
This is what I think:
1. If the individual is too busy to remember to clock out and in fact uses his or her lunch for lunch, then it would be okay to deduct the 30 minutes, otherwise that employee is essentially "stealing" money from the company by overstating amount of hours worked.
2. Now, if that same individual does indeed forget to clock out and does use his/her lunch period for work or work related matters, then the employer cannot simply deduct 30 minutes, because then the employer is trying to steal money from the emplpoyee.
3. The asking for a supervisor approval thing, to sign off on the fact the person did indeed work during his/her lunch hour is ridicilous. What if the supervisor is equally busy? What if the supervisor is absent? It really puts the burden on the employee to prove his/her case in and every single time. When I work during lunch, I just do not deduct it.
4. Now, if your company HAS a policy set out in your company HANDBOOK that states you MUST take a minimum 30 minute lunch break, which I suspect you do, then you should do it. The reason is, otherwise, what they would be doing could be argued as involuntary servitude (slave labor if you will for making you work all day without a solid break).
Go to the employment law section at freeadvice.com and see if anything pops up there. Also, try the Equal Employment Opportunity Commission (EEOC); this federal administrative agency may be able to better answer your specific question or at least point you to an individual or organization that could.
Hope this helps.