How does Square process card payments without requiring merchants to have traditional merchant accounts?

Absolutely Yes. I am surprised people have answered no. A whole industry existis that allows people like you (shop/business owners) to be able to process credit card transactions without requiring a merchant account. Its called 3PP (or 3rd Party Payment Processors).

Here are some examples:

www.2CheckOut.com
www.BrainTreePayments.com <--- Highly recommended
www.stripe.com
www.BluePay.com
www.CCBill.com
www.CCNow.com
www.CDGCommerce.com
www.FailSafePayments.com
www.FastSpring.com
checkout.Google.com <--- remember them!
www.NoChex.com
www.PayPal.com <--- yes, they too will provide you CC acceptance
BlueSnap (digital goods) (Plimus changed its name to Bluesnap)
www.rbsworldpay.com
www.regnow.com (digital goods)
www.epoch.com

A very comprehensive list can be seen here on a blog entry I did quite a few months ago. http://www.webscopia.com/2010/11...

If you search for "online credit card processors" or "online payment processors" you should be able to find quite a lot of them.

The basic premise is that the rate will be slightly higher than what you might be paying if you had your own marchant account. But you are not required to have a Merchant Account in order to be able to just accept credit card payments.

You make money, the payment processor who provide you with this facility makes a bit of money and your client can pay you using their credit card.

Everyone is happy.