There is a great post on Quora about who can/should buy Square:
Tier 1 – Clear fits if Square were for sale
Apple – Opportunity to vertically integrate the hardware and get even broader distribution as well as build a payments product with a “Apple” experience. Apple needs additional payments capabilities to do this (iTunes is a retailer of apps, not a payments system)
Google – Opportunity to reset Google Wallet and deeply integrate Ads / Offers into Square
PayPal – Opportunity to dramatically expand their offline strategy (now just a few large merchants with minimal usage) and cement PayPal as the Visa/MC competitor by tying PayPal accounts with “Pay With Square” Not a great buy vs. build tradeoff on Square’s basic card acquiring business.
Visa / MC – Big enough to pay up. Conflicts with some of their acquirers but that ship has already sailed for Visa with the Cybersource acquisition. Very valuable to control threat of Square disintermediating card payments over time.
Tier 2 – Plausible but a stretch
Amazon – Amazon loves low-margin disruptive businesses (bookselling, AWS) but probably would have joined this race already if they really wanted to.
Amex – Not able to pay a premium price now that it is a bank holding company and subject to bank holding company rules. Too dilutive to do a stock transaction.
Facebook – Has the ability to pay hasn’t been making big moves into payments, especially offline. Credits is a pretty limited use case.
Intuit – Strong strategic story but not a great buy vs. build fit given that it already has GoPayment.
Twitter – Consolidate Jack Dorsey’s companies into one. Increase commerce component of Twitter monetization. Would be an all-stock deal so difficult for boards of both sides to value the transaction.
Verifone – Great strategic fit, but very bad blood between the companies
Tier 3 – Huge stretch
Bank acquirers (Chase Paymentech, BofA Merchant Services) – Parent companies are [regulatory] capital constrained
FirstData – Great strategic fit however its a private equity owned and has tons of debt thus unable to pay
Monoline Acquirers (Vantiv, Heartland) – Great strategic fit, but not big enough to pay up for Square
Verizon/ATT/T-Mobile/Sprint – Already doing ISIS as a consortium. Not strategic enough for any one of them to buy alone.