Agreed, although I would word it stronger then that. If ANYONE calls about an employee other then maybe verifying that they actually are an employee, I have been told by every company I have every worked that you NEVER talk to outsiders about your employees, much less engage in illegal criminal conspiracies against the employee. If I ever got such a call, I would listen carefully, take notes, say nothing, and run up to Legal as soon as the call ended. Make it their problem. If it really the EEOC, Legal would likely call the FBI.
Forget EEOC. I would act the same way if the Girl Scouts called. Even is someone claiming to be IRS called. I have worked with IRS enough to know they do not act the way you describe. They send paper. Lots and lots of paper. Any supervisor stupid enough to do what you said would be fired for cause. Quickly.
Now small firms often do not know the law (or care about it) but any theory involving a small underfunded agency who cannot do it's normal workload in a timely manner is going to randomly go James Bond on someone is every bit as silly as it sounds.