Currently with the circuit that you have, you will power the Arduino through the output pin just for a moment when you hit the button. This could possibly hurt the Arduino.
If you run the black wire or negative power through the relay, instead of the red wire which is positive, the Arduino will be protected at least until it has power to it.
With the negative disconnected at the relay, the Arduino will not be able to see the button voltage so Arduino will not get hurt.