If the square wave frequency is fixed, a delay chip would do it.  

If the frequency is variable then you need to use a tricky thing:

Integrate the square wave in two capacitors, using two resistors, one
for each capacitor in a traditional R+C configuration.  

RC #1 would integrate some voltage and RC #2 would do the same but with
a voltage that is exactly hald the voltage from RC #1.  This is easy to
do with a double value capacitor or resistor.  

RC #2 (half voltage) should load a small capacitor via a diode. This
capacitor voltage would be almost precisely half of the RC#1 integrated
peak voltage.  With some exercises you can eliminate RC#2 and use a
voltage divider from RC#1. 

Using the comparator function of the operational amplifier or a LM339,
when RC#1 integrated voltage goes higher than the voltage on the small
capacitor loaded by RC#1 and the diode, it means the RC#1 integrating
voltage ramp is (more or less) in the middle time, so aprox at 90¡. 
When RC#1 deintegrating voltage ramp goes less than the small capacitor
voltage, it is aprox at 270¡, so the output of the comparator would be
your 90¡ delayed square pulse.

A small trimpot adjustment could be done to locate exactly the 90¡
position, and it should be almost the same for different frequencies.   

It should not work pretty well with square waves with dutty cycle
different than 50%.

Frequency changes should not influence the result of this circuit,
except for the RC constant, so the frequency should change inside a
specific range.  As more linear the RC#1 ramp, the better the accuracy
of the 90¡.

A microcontroller timming measuring circuit is the best solution for
that, so it can calculate the exact half up or down period of the square
wave.  But you have max frequencies based on the microcontroller
capacity of time measurements.
  
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Wagner Lipnharski - UST Research Inc. - Orlando, Florida
Forum and microcontroller web site:  http://www.ustr.net
Microcontrollers Survey:  http://www.ustr.net/tellme.htm