Rvijay: Shortbread and pressed cookies are sometimes hard to duplicate "exactly" when veganizing the recipe. For your cookie press you want a shortbread-type recipe - thats basically what they are, but usually with an egg to help them form/hold together better.
I would simply try the recipe you listed and sub. a vegan margerine for the butter and maybe either En-Er-Gee egg replacer OR smoothly blended silken tofu for the egg... both of those sub for eggs in cookies pretty well. In this recipe I don't think applesauce would be as good a substitute.
Good luck - sounds fun!!
Do a Google search for Vegan Pressed Cookies or Vegan Shortbread. Either type recipe should work.