Simple inline latching power switch
Probably like shown below.

Assume that the Arduino's I/O has gone low.

Q2 will then go into cutoff, and Q1's gate will be pulled up to BT1+ through R1.

That'll force Q1 into cutoff, disconnecting BT1 from the converter's input, killing both the converter and the Arduino. 

When that happens, the Arduino's output will be floating, and R2 is used to make sure  Q2's gate is held low until S1 is pressed.

When S1 is pressed, R1 and Q1's gate will be pulled low, turning Q1 ON, which will connect the converter's input to BT1.

That'll power up the converter, which will then supply power to the Arduino.

The Arduion's I/O will then go high, turning Q2 ON, which will connect Q1's gate to ground, latching Q1 ON (even after S1 is released) until the Arduino's I/O goes low, turning everything OFF AGAIN, which will start the cycle anew.