Why do square waves have this spectrum response?
The images below by Lucas V. Barbosa show the expected transformation from time domain to frequency domain.







A square wave should produce a series of odd harmonics decreasing in amplitude. 

Can you post a 'scope screen grab of the actual "square" wave? I suspect that it isn't quite square.



As shown in the update the sides aren't vertical so that's what your analyser is working with. As Vincente points out, if you can zoom out a bit we would expect to see another large spike - but not as tall as the 60 Hz ones at 180 Hz and a weaker one at 300 Hz, etc.





Figure 3. A typical harmonic analysis display. Source: PowerSight.

You might consider a better display layout if you are only interested in power harmonics. These typically show the fundamental as 100% reference and the harmonics as a percentage of that. Notice that the plot of Figure 3 has odd harmonics only and is probably caused by symmetrical wave distortion - possibly caused by rectifiers on a DC supply or switched-mode power supply.

Note that this display shows the fundamental frequency and RMS current at the bottom.