Do you need eggs for extruded pasta dough?
The original pasta recipes of old were simply wheat and water.  Eggs or oil (the two most popular "water substitutes" in pasta recipes) are not at all necessary to make pasta dough, but they will have an impact on how the dough turns out.

Now, it's hard to give a firm "this is the best way to do it" answer, because everyone has their own preferences on pasta doughs, but you will tend to find that egg-based pastas are a litte more "creamy" than water-based ones.  In my experience, they work fairly well for rolling and cutting, but seem to be a little to "sticky" when extruding.  When I've tried it, the dough has had some issues with sticking to the various extruder plates, causing the pasta to tear.

Now, I have not tried it myself, but I've heard that water-based pasta doughs to much better with extruders/presses, because you can control that "stickiness" a little better and make a slightly drier dough, which comes out much more smoothly.  I've been meaning to try it myself, but have not gotten around to it yet (I really love the flavor of egg pastas, so I tend to just make those and cut them, instead   :)   ).