Will a Tesla coil work with a square wave inverter output, used as input?
A &quot;high frequency&quot; if it is high enough will actually limit the current in the primary coil for 2 reasons.

The skin effect will increase the resistance of the coil as current will only be conducting in the surface of the coil. This will limit current at very high frequencies, and cause more losses as well. The skin effect will not be significant for a thin conductor at low frequencies say in the kHz region
The coil will have some inductance which causes an increase in reactance (impedance) at high frequencies - hence a reduction in current. Although, with only a few turns on the coil and no ferrite core the inductance will be very low - so you will need high frequencies for this to be significant.

Depending on the frequency of your square wave supply, the current levels through the primary coil might be acceptable. - but you need to make sure there is no DC voltage component to your AC supply. DC will draw lots of current through the primary.
As JRE has mentioned, the tesla coil needs to operate at its resonant frequency, which a slayer excitor does for you. Your AC supply won't be effective on its own unless it is tuned to the resonant frequency of your tesla coil - are you able to adjust the frequency to match your tesla coil?
Alternatively, if you add an interrupter with your AC supply, you will get inter-modulation which will shift the frequency of excitation into 2 frequencies, one above and one below the actual interruption frequency and this will make it hard for the interrupter to excite the coil at resonant frequency. Although you could still use this method by adjusting the interruption frequency up or down so that one of the modulated frequencies matches the resonance frequency, you will then suffer loss of efficiency as a result with half the power being lost to a non-resonant frequency.
From your question about grounding the secondary I assumed you were thinking of using it as an ordinary transformer. But I realise now this is not the case.
On a Tesla coil, the connection to ground is through the air, which is possible because the voltages exceed the breakdown voltage of air - and also due to parasitic capacitance with surrounding surfaces. If you add a grounding point your wiring might add too much parasitic capacitance which will suck all the energy out of a very high frequency, high-voltage, but very low current transformer. I don't think you need to change anything on the secondary coil, just get the drive for the primary right.
You could rectify the voltage from your AC source into DC and then use an interrupter circuit. Or possibly, adjust the frequency of the AC source to resonance - the disadvantage of this is that would probably be no feedback loop to automatically tune in to the resonant frequency, but it could work if you can tune close enough to resonance manually.