Businesses don't get to negotiate with Visa and Mastercard. They negotiate with banks. And banks negotiate with large volume customers. That's why McDonald's can afford to take credit for a $1 ice cream cone without losing 33% of it to the card company.
Square has a lot of options in the way they process transactions, and they have a very high volume (now more so than ever)
I'm guessing that they are going to try and get significantly lower fraud rates from pay with square (since it's a photo verification and not an insecure card strip) which can help push down rates.
They can afford to break even on the conventional credit card side of the business if they can make their margins on Pay with Square.
How does SB gets smaller fees through Square than negotiating directly with card providers?
How do card providers allow Square to sell their services cheaper than themselves?
And Square, you sure have a nice business, but the magnetic stripe is very insecure.