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.
Another is that Visa believes that Square is a threat, thinks they may be successful and has invested to mitigate that threat, get visibility into their activities, and so on.
You hope the iPhone or iPad doing the payment processing with the Square device is connected via network cable? I'm afraid I have some disturbing news for you.
I think some, but not all, will pay data plans. Square doesn't require a data plan, as evidenced by it being compatible with the iPod Touch, which has no radio:
> Requirements: Compatible with iPhone, iPod touch, and iPad. Requires iOS 4.0 or later.
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.