Butter can be replaced by margarine, at least by those who don't worry about trans-fatty acids in partially hydrogenated oils. The biggest problem, however, is eggs, which play a significant structural role in unyeasted baked goods. Leaving eggs out of yeasted breads is not a problem. The structure of yeasted bread depends on gluten, a protein which is plentiful in many wheat flours. When one kneads bread, one is developing the gluten, encouraging its proteins to extend and bind together to form a network of gluten sheets with pockets between them.
Yeasted bread rises because yeast gives off carbon dioxide as it metabolizes sugars in the dough.
This carbon dioxide fills the pockets between the sheets of gluten, causing the bread to inflate like a honeycomb of little balloons. This happens before the gluten has a chance to form. When one kneads the dough to develop the gluten, one flattens any gas pockets which might have formed. Also, chemical leavenings have a destructive effect on gluten, so one would be working at cross-purposes to begin with. Unyeasted breads are like yeasted ones in that they rise via a multitude of little gas pockets.