Prevent others finding me in Game Center

Have you tried contacting your credit card provider?

Seriously, Apple's intention is that iOS Game Center friends are meant for "real" friends—rather than "Facebook friends"—so exposing your real name is not a threat. If you look at the issue this way, it feels more of a security enhancement than a threat. You can be sure the friend you are friending is actually the person you believe them to be.

Moreover, if you are underaged and can't attach your own credit card with your Apple ID, it is, in contrast, more secure to have the user's (kiddie's) name obfuscated.

Finally some legalese from the Game Center TOS:

The full name associated with your Apple ID will also be shared with those users to whom you send, or from whom you accept, a friend request, such as in the Friends list, the Friends leaderboards, and the pending friends requests lists; and we may recommend games that you have played to your friends. If you prefer not to share your full name or profile with friends, you should remove that friend by tapping the “Unfriend” button at the bottom of that friend’s screen in Game Center.

For each friend's account you need to force an update, you should unfriend them and then establish the friend relationship again.

I'm thinking that the server just must be down. I'm having the same issues logging into my game center on my iphone 4s. Perhaps it will fix itself within the hour as things like this apparently have happened before.