Electrically it'll work, but mechanically it'll suck. Right now you hold the phone in one hand and swipe the card with the other. With this adapter, you'd need three hands: phone, reader, card. Or find a place to put the phone down while you run the operation.

For something like a card reader, you really want a device that plugs straight into the phone. If Square isn't able to build a Lightning reader just because Apple controls the port, that would be kind of irritating.

Is there a 'reverse Apple Pay' where one phone is the payment terminal, another phone is the customer, and they NFC exchange a payment?

Alas, no. There are third-party NFC terminals which work with iPhones, though, so you could at least do it that way. Square has one, which talks to the receiving phone over Bluetooth.