Getting square impulses from square wave input
You need a circuit which detects an edge (positive or negative) and generates a timed rectangular output pulse. To create a pulse which is 'in the middle' of the square wave you need both of them. The second one is triggered by the negative edge of the first one's output.  

There are many different ways to implement these functions, depending on what components you want to use and the characteristics of the input and output signals. Here's an example circuit that generates all three outputs, using bipolar transistors. The widths of the output pulses are determined by RC time constants C1*R1, C2*R2, and C3*R7. 







simulate this circuit &ndash; Schematic created using CircuitLab