Q: imessage/facetime login errors

I had this issue with FaceTime in the past, have you tried just re starting the device? That worked for me.

Get rid of whatever antivirus software you've got on it. I'm assuming that is what the 'shield' icon in the menu bar is for.

It can interfere with all kinds of things and may very well be blocking communication with Apple's servers.

If you get an error when trying to activate iMessage or FaceTime - Apple Support

having the same issue here, Mac Mini Server 2011. Tried fresh install of 10.11.6, and fresh install of 10.10.5, same issue with both. iMessage works perfectly fine on my other 2014 Mac Mini, my Macbook Air, my wife's Macbook Air, my brothers Macbook Air... etc. No idea why this isn't working

Call Apple Support.

I had this same problem for the past week or so. I tried everything I could think of, and every solution I found online. I reset my Apple ID password... even performed a clean OS X install on my MacBook to see if it was a hardware issue. Nothing worked. I finally contacted Apple support this morning... talked to an advisor for 15 minutes, then was escalated to a senior advisor. It's a known problem. The advisor escalated my case to an engineer and promised to follow up when the issue was resolved. I received an email an hour later, and voila... it worked.

Again, call Apple Support. My computer is no longer covered by Apple Care, and the call was free. The problem was solved within two hours.