Ive used both systems, trying the Eggbeaters when they first appeared after starting with SPD and for me, Ive found the SPD’s preferable. However, my experience relates to the first Eggbeaters and they may well have changed and / or improved them by now.
What I liked about the Eggbeaters was the ‘float’ or freedom of movement of the ankle/foot, which made pedalling very comfortable, so long as I was wearing hard soled shoes. They looked good, lightish and the four sided engagement option was really good and useful for me too.
This float is also what I wasn’t so keen on! I found I had to exaggerate the unclipping movement to disengage, that whilst I got used to doing quite quickly, I prefer a smaller movement for release, which the SPD’s provide. I felt more secure with the SPD’s so for those two factors, IIRC, I sold the Eggbeaters and now use Shimano pedals.