Either will do. I have a Uno with built-in Ethernet and a mega with separate shield. I believe both can use POE, I have never felt the need to use it with either.
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I'm amazed at what Tim wrote, and the price. If you want to go that way, I suspect you would be better off with an Etherten like mine. It has USB, Ethernet and SD card all-in-one and seems competitively priced. I guess an Arduino with no USB has a place in this world, but surely not for a school project.