I have MQTT supervisor installed under supervisor and it’s configured and working. I need to access this MQTT server from outside my network. I was hoping I could use my Home Assistant Cloud URL to do this (https://xxxxx.ui.nabu.casa). It doesn’t appear to work. Am I beating a dead horse try to get this to work? I assume port 1883/8883 are not allowed through those ui.nabu.casa addresses.
https://xxxxx.ui.nabu.casa/ runs as you stated in the cloud, while the mqtt broker runs locally.
You need to forward port 1883 on the router of your local network and acces it on the public ip name/address of the router 
Yeah, I guessed that was going to be the case.