In the settings, there's a way to shutoff automatic updates, which I set. But I don't trust post-Corina HP at all and I wanted to firewall it from the internet.
It turns out my D-Link router is vastly more annoying than my printer. Unclear setting menus, no help or manual, and settings that appear to work will block every computer from the internet when the router is reset.
I hate the state of modern consumer electronics.