Can I get a square-wave inverter as source for my laptop charger?
Probably yes.

The laptop power supplies (they are not actually "charger", the charger is inside the laptop) are switching power supplies and they first rectify the input AC voltage, then process it to lower DC voltage. The input rectifier can work with square wave form even better than with sinusoidal.

Completely another talk is why you need to double convert the voltage from 12V to 220V and back to some low (what?) voltage. 

Most laptops with 3 element Li-ion are powered with 12V external voltage, so you simply can power your laptop directly (well, if it has 3-element battery). Check the laptop power supply output characteristics for that.

Another, even better solution (suggested by Dan D. in the comment) is to use dedicated laptop car power supply (often wrongly called "charger"). There are many types of these and they can power laptops with any battery configuration. The efficiency of such a configuration will be higher than using 220V power supply with inverter.