I don’t work for Square so I can’t talk about their specific implementation, but here’s how it should work:
When a “tab” is set up, the app can register with iOS to receive notifications when a user enters the geofenced region. As usual, the app will be suspended when you close it (i.e. not running), but when iOS detects you’ve entered the region, it’ll move the app into the background so it can run temporarily. In the case of Square, they might get a better GPS fix on you and tell their servers that you’re actually really close to where you have a tab open.