This might be what you are looking for (assuming you are still trying to solve this problem). This is stevenvh's answer drawn out as a schematic, and simplified by removing the AND gate.
Rather than use two sin wave sources, here I am using a capacitor and resistor to create a little less than 90º phase shift. I'm not sure exactly how accurate you need this. Alternatively, you can just use the 90º phase shift voltage source you showed in your schematic.
These two signals are turned into phase shifted square waves by the comparators. If you use comparators with open-drain outputs, then you don't need the AND gate, as the outputs are basically half an AND gate each. You just need to add a pull-up resistor, and you're done.
- You will need comparators with an open-drain output.
- There is no explicit hysteresis in this design. Either use a comparator with internal hysteresis, or add some.
- If you really the output to be perfect, then you need exactly 90º phase shift, so you'll have to use the opamp differentiator that stevenvh mentioned in the previous question.