In my experience, Delta Sky Club, Admirals Club and United Club are comparable to each other. But I know that some travelers have strong preference toward one club or another. All of them are much more comfortable than publicly available halls: there are quiet rooms in the clubs, they offer snacks, drinks, and free wi-fi. However, they don't have beds.
At ORD, three main terminals are connected to each other on the airside. So you can go to any of these three lounges without leaving the secure area (the International Terminal is not connected to Terminals 1,2, and 3. But all domestic and most international flights by US carriers depart from Terminals 1, 2, and 3). United Club will be the most convenient for you: it will be the closest to your departure gate, United staff will inform you when the boarding starts and assist you if your flight is delayed.
As DJClayworth suggested, you can also book a room in the Hilton Airport Hotel. It is connected to Terminals 1,2 and 3 by an underground tunnel. (The same tunnel connects terminals to the subway station and parking garages.)