Why is a square wave aliased?
In addition to @hotpaw2 explanation, a graphic. There are two analog square waves (red and green), with different lengths. They are depicted with a fine sampling, denoted by crosses. Their actual sampling is denoted by circles. The  red one is shorter than the green one, as can be seen in the interval $]0.7\;0.8[$. Yet, the  sample points are the same. Thus, from the circled samples, you cannot tell one from the other. This  would happen for any square wave whose end falls inside the  blue segment. You can see here the loss of information.



With the  most simple schemes  (using cardinal sines), these different sampled squares would be reconstructed as a single analog signal (as their  discrete samples are the same.) Depending on the  precision on the sampling, several analog-digital-analog signals can be theoretically recovered,  possessing a certain amount of ripples, or overshoot, or Gibbs phenomena.



You can reduce the  ripples in the interior of the square wave (as seen going from blue to red to green), but the amplitude of the overshoot will remain constant  at the  vicinity of the  edges.

As commented by @robert bristow-johnson, there are a number of alternative techniques. The document Alias-Free Digital Synthesis of Classic Analog Waveforms (Stilson and Smith) provides some of them. I add a recent technique, based on Finite-Rate-Innovation: Feasibility of {FRI}-based square-wave reconstruction with quantization error and integrator noise, B. He et al., ICASSP 2015.

Finally, for your last question "Why does superposition hold on such a simple construct?" This can be regarded as a problem of approximating a non-continuous function on a basis on continuous ones: you can get pointwise convergence (all samples converge), but no uniform convergence (not all converge at the same rate).