TLDR; Just give me the new fix
This made things work for me when put together with "Stuff Everyone Has Already Done" down below. Uncheck Use TLS 1.1 and Use TLS 1.2 in Java Control Panel, Advanced Tab:
Be sure and close all Firefox windows and restart the browser so that change takes effect.
Also, make sure and run the remote console twice (to the same ILO server) before you consider it not fixed. I'll explain that 2x run thing down below, but essentially on the first run, the remcons class jar file gets cached but NOT used and the next run it actually gets used. For some reason it will only run "from cache", if that makes any sense. It's like old routers that could run from RAM vs run from FLASH, in this case run from CACHE :)