... look bad, and to answer a question about what you've done in your job may reflect badly on them, simply concentrate on the procedural details of how you solved the problem, but say, politely, that the nature of the problem is confidential, and you don't want to unwittingly harm your current employer's reputation.
That's something the potential employer would be likely to respect--they'd want to be treated the same way.
You might also want to do some research beforehand on the potential employer. It's not uncommon for someone contemplating a takeover to use employment interviews to gain information about a company.
In the interview, if it seems the questioning is a bit too pointed, that the questions seem to hover around problems of the company, it might be best to simply say, sorry, I really can't speak to specific problems inside the company--I'm not authorized to do that.
Besides, the best answer to any question about problems is what you've done to prevent them.

Cheers.