Its good that some people asks for specific suggestions. Having the right shoes can make all the difference in the effectiveness of your workout. I would say focus on flexibility, stability, and overall style. Try Asics, I've used them a few times at the gym now and I'm really impressed. They breath nicely and afterwards my feet can't even tell I've been working out. They do seem to run about a half size too big, but it hasn't affected the comfort for me at all. There are some negative reviews in regards to these shoes holding up to running outside. It's probably fair to mention that I don't think these shoes are really meant for outdoors, they're cross training/gym shoes, which means exclusively indoors. That's probably why these are quite a bit cheaper than running shoes as well. Most gyms, including mine, require you to use shoes that haven't been used outdoors anyway so as long as you are respectful of that policy this shouldn't be an issue. But if you're looking for something to use both indoors and outdoors, this might not be the shoe for you.
The statement in bold seems hard to believe. More likely, most gyms only require that the footwear is athletic and closed-toe.