Square and Stripe fit very different markets, there's some overlap in both could be used to power eCommerce sites but Square mostly targets traditional merchants selling products and services, Stripe targets developers looking to add payments to their product (SaaS, online marketplaces, etc.)

Either way, this is a perfect fit for Square - they already offer simple "one-page" web stores for merchants, tying in a full-featured website or store instead makes a lot of sense.