How do I design a circuit to convert a 100 mV and 50 Hz sine wave to a square wave?
The easiest way to do this would be to use a comparator. 



Picture taken from linked site

All you have to do is set your Vref level to where you want your square wave to trigger. When the sine wave crosses the Vref level, the comparator output will go high. As it approaches it again and goes below the Vref level, the comparator output goes low.

You will then get yourself a square wave. 

Be aware the example shown in the graphic is of a non-inverting comparator. An inverting comparator works with the same principle, but the output is inverted