What are the effects of powering AC devices from a square wave?
AC-mains appliances are engineered to present a certain impedance (load) for the specific 50/60Hz mains frequency.  Feeding them a square wave of the same (fundamental) frequency is - mathematically provably - the same thing as feeding them not just a sine wave of 50/60Hz, but super-imposed on that also a 150/180Hz at 1/3 the amplitude, and a 250/350Hz sine wave at 1/5th the amplitude, and a 350/420Hz sine wave at 1/7th the amplitude (i.e. all the odd harmonics at decreasing amplitudes) and so on, in theory up to infinite frequency, but in practise with domestic wiring the losses &amp; impedances in mains distribution networks &amp; appliances, they peter out at kHz or maybe 10s of kHz (for a 50/60Hz fundamental).  These higher frequencies are the result of the high rise-time &amp; fall-time of a square wave.

For some appliances this doesn't matter so much - like resistive loads (e.g. heater elements).  For inductive loads (e.g. motors) or some capacitive loads, that typically means a significant proportion of the energy you're hoping to deliver into the appliance won't actually be used for 'work', but will instead be dissipated as heat, because the inductive/capacitive load is tuned for 50/60Hz, not the higher frequencies.  So, depending on the specifics of the appliance, powering them from a "square-wave inverter" might work ok, might not, or might even damage it permanently, it all depends on the specifics of the appliance.  Even low power switch-mode power supplies that in one respect would be more impervious to this kind of treatment, actually aren't, and will in fact deliver poorer (noisier) DC to their electronic circuits, hence gadgets "play up".

The short story is, the DC-to-AC inverter industry for the last few decades has gone through a maturation process of inverter designs to overcome this severe shortcoming of square-wave inverters, first with 'modified sine wave' (a misnomer, actually a 'modified square wave' where there's an off-period on either side of the 'zero crossing'; and then on to "pure sine wave" inverters, which also aren't really sine waves.  Some filtering with inductors &amp; capacitors can also help to slow down the fast transitions of square-wave, but because of the low frequencies involved, they're large components &amp; therefore expensive.  Modern 'sine wave' inverters create a bi-polar wave via a modulated PWM at a much higher frequency (say, a few kHz or even 10s of kHz), which can be filtered out to achieve something more resembling a sine-wave (at least under load).