square wave skin depth
Skin depth is defined only for sinusoidal fields and depends on frequency. 

A rectangular signal can be seen as a series of sinusoidals signals with frequencies \$f(2n +1)\$ with \$n = 0, 1, 2, ...\$ where \$f\$ is the frequency of the rectangular signal (cf. TonyM's comment). 

The higher frequency components will have lower skin depths. Higher frequencies will be attenuated more than lower frequencies. I.e. the waveform inside the conductor will not only get weaker it will also change its waveform.

So to answer your question you first have to define skin depth for non-sinusoidal fields (e.g.: is it where the amplitude of the new waveform is \$1/e\$ of the amplitude of the original waveform; or is it where the RMS of the new waveform is 1/e of the RMS of the original waveform; ...)

So only when you have a definiton of skin depth for non-sinusoidal signals your question can be answered.