I think it has something to do with Jennifer Lawrence's iCloud being hacked and the nude pictures published. Which apparently I live in a bubble. I didn't know it happened till today and didn't know about 4chan till today.
If by stop working you mean you're getting a message to call Apple Support and give them the Code N0. from the error message, please go right ahead and give them a call. All that they will ask for is your Hack's serial N0. I promise you will not be frowned upon for using a non apple machine and they will fix your iMessage login problem within 2 minutes. As a matter of fact, you will get a nice little call back asking if you were happy with their service in solving your problem etc etc. The latest outrage with the iMessage service, I think is to do with the security breach with iCloud a month or so ago and apple making a few security changes which affected some of us on non apple machines. So there is nothing you can do at your end to solve it, so go ahead and give them a call and its a free phone number too. Good luck.
Had this issue, and called Apple. They couldn't find me on their systems with the serial number my system's using, but got there in the end with other details they had for me on my iMessage account. After that they fixed it very quickly.
All in took about ten minutes (including the time Apple's voice recognition took to give up trying to understand my thick home-counties accent and dump me in a queue).