Generally, I work out in my running shoes. However, that's only because I like to run before I lift. The problem is that doing anything other than running in your running shoes is pretty bad for them - especially lifting, as it messes up the stress/pressure patterns on your shoe so it can end up feeling off while you're running. However, for most of us that aren't running for any reason other than to stay reasonably fit, that doesn't really matter.
I find that I get totally fine support from my running shoes when i work out - but then again, if I were to start squatting some serious material, or doing intense Olympic lifts, I would definitely not be wearing running shoes. However, for doing the average person's session, you should be fine. Really, it all depends on whether or not you feel like you're going to break an ankle when you're squatting - in which case you probably want some running shoes with better support anyway; or if you really care about your running shoes.
That being said, if I'm going to the gym and I'm not planning on running, I just wear flats. Nike court traditions (they're velcro and my ankles haven't snapped yet). I have a friend who loves his Nike frees, but I've never so much as tried a pair on.