First, Square might avoid needing a traditional credit card with their products somehow. I realize mobile payments and other methods have been tried before but eventually it's going to work. This would have the effect that Craigslist and Ebay had on newspaper classifieds by reducing the market for VISA's products for banks and merchants.

Second, Square can definitely compete in the cardspace. What's impossible for them now could change with a big enough operation.

Third, Square can use their ubiquitousness to lower fee structure as has been said. That will reduce market share of the fees in the payments 'stack' only; VISA/MC will still exist everywhere. I think this will be necessary for them because their biggest competitor right now is cash, because they want to do every financial transaction in the world.

Since all of these 'disruptions' would probably boost the other I predict a combination from them from the company that gets as big as Square wants to be.