If you want to build it yourself you'd be better-off making a high-power oscillator, and then you can switch it on & off with the Arduino (using a relay or MOSFET, etc.) if you want to control it with the Arduino.
I have one of those mist generators but it's a pre-made sealed unit. The power supply for mine is 12V at about 4 Amps (I think) so that's about 50 Watts and I would assume it's cheaper to buy a pre-made mist generator than to make a 50W oscillator.
Plus, assuming it's generating a 1.6MHz square wave, that could generate lots of radio frequency interference if it's not well shielded.