Hi Sandra! According to the official "Features and Device Compatibility" page on the Square web site, both Square Register and Square Wallet are compatible with Google Android devices running 2.3 or higher. Since the Motorola Moto G runs Android 4.3 by default, then it falls into the compatible Android phone category.
There are other requirements, and these are:
- A 3.5mm headset jack with microphone (audio jack adaptors are not supported)
- A screen size specified as Normal or Large (Small screen sizes are not supported)
- Location services enabled
- Access to Google Play
The Motorola Moto G has all these requirements as well. So the answer to your question is yes, it can be used with a Square credit card reader, specifically through its built-in 3.5mm headphone jack, and it can be used with Square-related services as well.
Good luck. Hope I was helpful.