This was a totally inappropriate request which I would have immediately objected to an would have reported to a supervisor right then and there. You should nevertheless report this behavior to a supervisor as soon as possible regardless of the outcome.
I agree that this was inappropriate. You should write a letter to the supervisor and the Director of the office. I would also consider submitting a request to the USCIS Ombudsman's office because this is behavior they need to know about and will hopefully respond to. You can submit a report online.
Your objection sounds bizarre. Why is it such a problem for you that it warrants righteous indignation and spending time posting on AVVO asking about the legal underpinning for the request?
The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter,not should it be viewed as establishing an attorney client relationship of any kind.
Such a request is completely inappropriate and should have been brought to a supervisor's attention immediately. Having an experienced attorney at your interview would have prevented this shenanigans and make sure the interview was properly conducted. If your case was not approved you should retain counsel without delay.
While this answer is provided by a Florida Bar Certified Expert in Immigration and Nationality Law, it is for general information purposes only and an attorney/client relationship is neither intended nor created. You should seek out qualified counsel to review your case and provide you with advice specific to your situation. Review Mr. Devore's Avvo Profile for more information about his expertise in immigration law and how to contact him to discuss your case.