How do you know it's a square wave? Though most inverters aren't perfect, it should be an approximation of a sine wave unless it's the jankiest inverter on the planet ;)
There's no real issue going from DC->AC->DC again. There are efficiency losses in the form of heat in the inverter, and the normal heat in the charger.
Check the current that your 12V socket is capable of. Most are 10 or 15 Amps which means you get 120-180 watts. Depending on how the charger works, I expect one of three things to happen. The charger works and happily charges your board, the charger draws too much current and the inverter gives off a low voltage beep, or the fuse in your car supplying 12V blows. Aside from a 50 cent fuse, nothing should break. Give it a try!