If they become the new merchant-interface, there's nothing stopping them from offering an alternate payment mechanism that doesn't rely on credit or debit cards.
I'm an engineer at Square, so I can neither confirm or deny such speculation ...
but in an interview with Wired, CEO Jack said
"People understand credit cards. They know exactly what to expect from them. They’re used everywhere. But Square is payment-device agnostic. It doesn’t matter how you pay. Some of our users accept only cash."
http://www.wired.com/magazine/2011/05/mf_qadorsey/all/1
Building the technology that could support Visa/MC/Amex alternatives and the userbase that gives it critical mass is probably the best way to get bought out by Visa/MC/Amex at an inflated price. No one says they actually have to ship their alternative payment system.
If they become the new merchant-interface, there's nothing stopping them from offering an alternate payment mechanism that doesn't rely on credit or debit cards.