Arduino battery power stay on for button - Not quite working
Your DC-DC converter is not isolated, and it places the whole supply at its output when its negative input terminal is floating. 

This means that the Arduino has the full input supply from its Vdd and Gnd terminals to the 330ohm resistor on its output and from there it should turn on the mosfet. 

If you have Gnd otherwise connected to the Arduino, then you have 19V across it. 

You should use a high-side switch, or a converter with an enable/shutdown pin instead.