You need to know what the specific battery is capable of, when using a mech. Use ohm's law to calculate what the amperage will be for the atomizer resistance and battery voltage, and make sure that it falls comfortably within the battery's safe operating parameters. You should use
batteries that have a high maximum output rating, such as AW IMR cells, which are safer chemistry than typical Li-ions.
That said, 2 ohms should not be a problem for most
batteries, but some have come here asking about oddball
batteries that were well underrated for that. Do your homework and get proper batteries.
Switches generally have two important factors in their rating: current capacity and breaking capacity. The current and voltage used affects both, to an extent. As long as you stay below both the current and voltage listed, you should be fine.
That said, DC current is typically harder to break than AC current, so I would limit the DC voltage to
much less than the rated AC voltage, if no DC specs are listed. Also, the higher the voltage, the harder to break the circuit. That's why the switch is rated for less amperage at 250v than at 125v, because the higher currents at higher voltages tend to arc longer.
I'd say for a 4.2vdc, 2 ohm circuit (4.2/2= 2.1 amps,) that switch would probably be fine. I'd look for switches with DC voltage ratings in the future, if that one wears out quickly.