For the underqualified problem, it's all a matter of demonstrating transferable skills. Of course companies like to see people who have worked in the same field or in the same function before since it's easier for them to get up to speed and start contributing to the company quicker. But even in this environment, at least in the healthcare field where I'm recruiting, all the companies I've spoken to have said they're very open to hiring career switchers if they have the intangibles. As one recruiter put it: It's easier to teach someone about the industry than to teach them to be smarter or a leader.
As for your overqualified problem, that's probably not a problem either. It's possible if you go after a company that generally doesn't hire MBAs it's possible that they may view an internship as being similar to an undergrad internship, or may offer an MBA an entry-level position, but I think these are in the vast minority of companies you're going to recruit for. Especially the employers who recruit on-campus are very familiar with MBA hiring, and their internship and full-time hiring positions are very different for MBAs than for undergrads. As an example in CPG marketing an undergrad may get hired as a brand analyst, while an MBA would be hired as an associate brand manager or even in some cases a brand manager, or in consulting or banking an undergrad would be hired as an analyst while an MBA would be hired as an associate.