Hi,
Try these things separately:-
1) NVRAM Reset How to Reset NVRAM on your Mac - Apple Support
2) Safe Boot
Both clear certain caches that are not touched even in a regular reboot.
If that does not seem to work then check you can see the Serial Number of the Mac in "About this Mac" from the Apple Icon Menu top left.
In EL Capitan it is in the Overview tab.
If this is not there then the iMessages account will not log in.
It can be missing if you have had the Logic/Mother board replaced and someone has not "flashed" the number back to the new board.
You will need to either visit an Apple Store or the place you got it repaired to have it put back.
If 1) and 2) have not worked and the Serial Number is there then you will need to call Apple Support.
Explain you have an Apple ID issue (free to fix).
You will need to tell them it is the Mac's use of the ID with just the iMessages servers.
The Level one responders are generally using a script to help in Apple Care cases.
Most likely you will have to ask to speak to someone higher.
Level 2s are still likely to get you to test many settings and things and possible do Data grabs and involve the Engineers.
Politely insist that you want the Server reset in relation to your Mac's use of the ID.
At this point they will probably tell you it will take 24 hours (most people seem to report a "fix" overnight)
You DO NOT need to do a Reinstall or an Update to "fix" this.

7:21 pm Thursday; March 31, 2016
iMac 2.5Ghz i5 2011 (El Capitan)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad