Messages having issues logging on recently on Yosemite?
In this order, try this:

Sign out of iMessage on all devices (if you want to be lazy, just the computer)

Remove iCloud from the computer (in the Accounts tab -- don't worry, we'll be putting it right back on)

Restart the machine

Repeat in reverse (add your iCloud account back, and then sign into iMessage).

That's basically what AppleCare would have you do.