If you are referring to the App Button, no… but if properly coded, then yes a sketch and physical button should keep running even if the device is not connected to the server at the time
I am not understanding why your Arduino keeps shutting off, nor why it will not automatically reconnect when powered on. EDIT - oh, you are using BT/BLE OK, that is “normal” for now I guess)
Check out this connection example… EDIT - However, it is geared for WiFi use…