I just recently switched over to Square and I'm using the Magstripe reader. With this reader the Square App will function in "Offline Mode" where it will accept charges but won't process them until you reconnect to wi-fi. If you're on LTE it works in Normal Mode just fine.
I'm running a small BBQ food truck business and this is pretty convenient for me overall. Simple to use, works well, no issues so far (about a months worth of use), and the money gets deposited the next business day (so Friday's sales post on Monday morning).
I'm using an old LG cellphone to run the Magstripe reader and so far all is well. Before Square I was using a company called SpotOn and they offer a ton of services, along with credit processing, but they were pretty shady. The fees were all over the place depending on the type of credit card and it was averaging over 4% of sales for "interchange." On top of that it was $35 per month...Square is 2.75% for every swipe so it's quite a bit better.
Note : I bought the Bluetooth chip reader but it doesn't work on my phone...lucky enough the Magstripe reader comes with chip reader, and that works just fine.