So the red+black wires connected next to the switch are the 5V power supply, and the other 2 wires to go the atomiser module?
If so, you should measure the voltages around the switch relative to the black/psu-ground. Otherwise you could be measuring meaningless floating voltages against each other.
I suspect that you can simply connect an Arduino pin directly to one of the switch contacts to control it. If course, the Arduino and the board must share a common ground. To simulate pressing the button, the Arduino sketch can change the pin to OUTPUT & LOW for a couple hundred milliseconds, then back to INPUT.