For your purposes of Square, I think that an iPad 1 will be just as good as an iPad 2 for your needs. (I’ve owned both the iPad 1 and the iPad 2.) I do trust Apple with refurb products.
 
However, I noticed that both the iPads you were considering were WiFi only.  Unless you’re confident that you can provide WiFi from your laptop, I’d seriously consider the iPad with 3G connectivity. Being able to connect to the Internet from anywhere would make a big difference – particularly for using Square.