I know that you are sure of the EGI being the problem (and this tread is a lil old) but I had the same problem last week with my 92 S13 HB. It ended up just being a grounding issue, if you want to check it simply just hook up jumper cables to your car battery normally and then connect the other ground connector to the body of your car (if this dont work connect the ground to your engine) and try to start it. *** Warning *** DO NOT let the power cable connect with your car as this could cause cool but unwanted effects to your car.
If this works simply run a wire from your body to your engine (0" gauge should do it) and BAM! its fixed!! I hope this helps as it did for me! Saved me like $25
a EGI Relay!