Excellent question KBS.
You can substitute eggs with water, however I wouldn't advise it. Eggs besides providing a liquid to bind the pasta, provide flavor, protein, and fat. Noodles made with eggs will taste different and have a softer texture then ones made with water alone.
With that said, you can make the pasta without them (lots of dried noodles are not made with eggs). I would substitute half of the all purpose flour for semolina flour, a coarsely, ground flour made out of durum wheat. I would also increase the amount of olive oil to 1 Tbs. I"m not sure how much water to add...add it slowly to the flour until the dough begins to come together and then knead, roll out, cut, and boil.
In reference to the Pasta!!!! recipe! If you don't want to use eggs or don't have eggs handy, is there an alternative? And what will be 'missing' by not using the eggs?