Hi,
There are several ways that iMessages shows it is not connected to the iMessages Servers.
1) It will not Login when you Open the App or change the Apple ID.
2) it will turn every Contact you try red and say they are "Not Registered with iMessages" (when other devices can iMessage them)
3) you get are far as typing the messages but each time it say "Not Delivered" and shows an Exclamation mark in red.
There are several things you can try:-
Eric's List.
Checking that your Serial Number shows in "About this Mac" (Apple icon Menu top left) in the Overview tab
Trying a NVRAM Reset How to Reset NVRAM on your Mac - Apple Support
Trying a Safe Boot
If none of these work - in turn and restarting Messages each time - then you need to contact Apple Support (Same as Apple Care).
This can still be long winded as you have to speak to the Level 1 responders first and explain you have an Apple ID issue (Free to fix).
The Level 1 responders are script led and it can take a while to get to someone higher.
Level 2s may know about the issue (Apple are aware and trying to find the cause but the numbers are relatively small and there is no apparent trigger).
However this leads to more data gathering and more tests of your set up.
Reinstalling, Updating or even Upgrading do not work.
Ask them to reset the iMessages Server as far as you Mac's use of the Apple ID is concerned.
As the Serial Number is used is allows the iMessages Login to be Devices Specific.
This means they can tell which login is causing the issue.
It seems that Level 2s can no longer reset the iMessage Server.
This will need the engineers (judging by other people who have posted back) and may take up to 5 days (people have been told anything between 24 hours to 5 days)

8:47 pm Saturday; August 13, 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),
iPhone and an iPad (2)