For the sake of the future generations:
the way DNS resolver at Microsoft works is that the hosts login.microsoftonline.com and drivethruclient.azurecr.io depend on the accessibility of the hosts *.msidentity.com, *.trafficmanager.net, *.cloudapp.net, *.cloudapp.azure.com, *.azcr.io . May be there is more, but this is what I have seen so far. You can see the CDN at work in this ping, for example
ping login.microsoftonline.com
PING www.tm.ak.prd.aadg.trafficmanager.net (20.190.159.3) 56(84) bytes of data.
If a firewall blocks trafficmanager.net az login will fail. DNS resolver rotates the (CDN?) providers every 5-10 minutes. The actual dependency can be r1007eus.eastus.cloudapp.azure.com, etc. This explains occasional failures of of the az login for devices installed behind DNS filters. This means that whitelists 1) are not complete 2) will never be complete. Using a custom domain in Azure Container Registry is a private preview feature, and I am not confident that it is not just another redirect. A typical example is Cisco Umbrella in the "Allowed only mode" when the DNS filter blocks access to all destinations except those specifically allowed.
In my view this is a deficiency of the Azure container registry limiting the deployment options. If you are to deploy docker container behind a whitelist only firewall consider using a different docker registry, or a VPN.