It's not a network problem. Your hard drive has the PlayOnline files "corrupted". You can invoke the disc install menu using one of the functions on the pre-login screen. Have the PlayOnline install removed and install again.
It's very unfortunate but PlayOnline files on XBOX360 are signed with the console local certificate and if you had your console replaced at any point, the files are no longer valid (for the current console) and need to be removed. A fresh install is due for you unfortunately.