Its hard to give you a specific laptop. All I can say it try online websites. I bought mind from NewEgg.com. And they have a large selection and give you good details. If you want to play games on a laptop, do not expect to get good performance while using a battery. And you probably want duel core (do not think you get get a quad core laptop for 600 bucks), any ATI 4500+ should be able to run games.
I would say no to that laptop you were considering. Reasons: because it is a i2 CPU, and it is refurbished. With 600 bucks you can probably buy a better new laptop, or even a better refurbished laptop.