How would Square switch to having accounts?
Or a "wallet", if that is a better term for it. These are simply accounts that carry a balance, just like what PayPal, Venmo, Dwolla, American Express' Serve, etc. all have.
Whose accounts? The customer's or the merchants?
I think you are confusing what I am saying with the old "merchant"-centric models. Square is P2P, except one side doesn't necessarily need to have a square account -- just a magstripe card.
Every person running the Square app can be a merchant though. So there really isn't the concept of "customer" and "merchant" anymore -- at least not to Square (and Dwolla, and Venmo, and PayPal, and Serve, and ... ).
And just to clarify, that person who just swipes a magstripe card doesn't have an "account" with Square.
How would funds get into those accounts?
Same ways as with PayPal, Dwolla, Serve, Venmo, etc. One way is to receive a payment from someone else and just hold onto it rather than having it simply pass through and funds withdrawn to a bank account. Or the funds can be add using an ACH pull, just like how PayPal, Dwolla, VenMo, Serve, etc. do it.
It's the natural next step for Square to add a wallet once they have enough people using their app. I'm not sure why they haven't done it already. Probably because it adds complexity and is expensive to administer. But eventually they will, I would bet.
Starting out Square Register by having it be the simplest it could be (a free app for iPad + a $10 dongle) and making the design super simple yet be a fully functional point-of-sale system is a very smart and unprecedented approach.