This question can be taken two ways really:
* Sending analog audio on a "digital" wire. Everything is analog in a wire. The digital part of a digital signal is that the sender drags the line to either logic high or logic low and the receiver tries to figure out whether the signal is closer to high or low. I'm not sure why you would want to send analog audio over an ethernet cable, but there's nothing stopping you but electronics ability and common sense.
* Sending digital audio. This is also possible. You just put the DAC/Amp in the headphone like with Bluetooth and noise cancelling headphones. You essentially get a Bluetooth headphone, but wired, and only compatible with whatever frankenstein setup for streaming you have. Specifically for ethernet, you could just use a similar microcontroller to DLNA/Cast/AirPlay speakers, plug your ungodly creation into your home router, and stream audio from your phone, over wifi, to the router, over ethernet, to your headphones with your app of choice. Again, the only thing stopping you is your handiness with electronics and common sense.