There are no Zigbee Router devices that communicate over Ethernet. Zigbee Routers always communicate with each other, Zigbee end devices, and the Zigbee controller via the 2.4GHz Zigbee protocol.
You could use PoE to simply power a Zigbee Router, but that doesn’t really solve anything.
You could use a PoE Zigbee Controller (e.g. like The Tube’s devices) which would create its own new Zigbee mesh network for an outbuilding. HA would communicate via Ethernet to the Zigbee controller, and would need to run either ZHA or Zigbee2MQTT (Z2M). If you already have a ZHA based Zigbee controller, then I would recommend using Z2M on the new controller.
This is really the same idea as what I originally suggested regarding simply using a Philips Hue bridge in the remote location. You could easily pair Zigbee lights with the Hue Bridge, and then have the Hue bridge integrated with HA via Ethernet. This is a very easy and quick way to add another Zigbee mesh network to your HA system.