Here is what is going on. Ever since iOS 4(?) there has been a feature where if you connect to a WiFi network it will try to ping this site:
http://www.apple.com/library/test/success.html
If it is a success then it lets you connect, if it is not a success the phone trys to bring up what would be a Login page for corporate wifi networks. Example, public access points that have a login page, etc.
It seems that some moron at Apple deleted the file located at
http://www.apple.com/library/test/success.html and that is why it is not working.
This is not an iOS 6 issue but rather a problem with Apple's website.
Rebooting your phone will not help, nothing will help until Apple fixes the problem on their end.