Can the Beaglebone handle Power over Ethernet?
Imho no. Would be a gread feature!
As a matter of fact, no it cannot support it.
That is not a feature included on the board. You might be able to use
something like  to break-out the power, but you would need to make
sure it is a 5V supply to make it work, not a typical PoE supply. You
will fry your board if you don't make sure the voltage is 5V.