What is the square symbol on the Wii U gamepad?
Solution 1:
That is the location of the NFC reader/writer built into the gamepad (see page 8):
NFC (near-field communication) location
Built-in NFC feature allows the system to read or write to an enabled card or other device.
This works with Wii U software exclusively designed to use this feature (see game packaging for compatibility information)
To the best of my knowledge, Pokémon Rumble U is currently the only game with any support for that feature.