It is not possible to use the Yun with Breakout server via the network port, because Breakout Server only bridges the serial port connection of an Arduino to the web via Websockets. However it is possible to run Breakout on a Yun without any computer involved (other than pushing the code to the Yun). I haven’t tried this in a few years so I’m not sure if these instructions work (and note the warning about potentially bricking the Yun): https://github.com/soundanalogous/yunBreakout.