From a technical perspective there's nothing that would prevent you to implement a payment terminal using the built-in NFC reader capabilities. All Android NFC devices are capable of speaking the reader-side of the ISO/IEC 14443 contactless smartcard protocol (as used by EMV payment cards).

However, there are a few other things that may inhibt such efforts:

• If you want to use a ready-made (and certified) EMV kernel (i.e. the reader application), you might not be able to get one (either because manufacturers do not target Android, or because manufacturers fear the risk that their proprietary software is extracted from the device).
• It's unlikey that an Android device (unless its sole purpose is being a payment terminal) would get certifications necessary for payment terminals (mainly because other software could bring transaction data, typed PINs, etc. at risk but also because many Android devices lack suitable protection features on the hardware level). Without those certifications, it's unlikely that you would be able to enrol your terminal with an acquirer that would process payment transactions received through the mobile phone.
• Eventhough the NFC hardware supports reading ISO/IEC 14443 contactless smartcards, many Android devices are not too good at that. This is mainly due to bad* antenna design. (* actually not necessarily bad design but design goals targeting simple NFC tags and no contactless smartcards.)

I personally think that the latter two were the main reasons for Square to provide their own device. Though this is just pure speculation and not quoting from any official sources.