Hi, Anna,
You can leave the skins on zucchini, no problem. Eggplant can go either way. Sometimes I find the skins a little tough or bitter, especially on the larger ones, so I usually take them off. It's just a matter of preference, though, they're perfectly edible.
Also, eggplant can release a lot of water when cooked, so in recipes where this would be a problem, you'll often see a recommendation to sprinkle the eggplant with salt (once you've cut it up) and let it sit for a while. (I think about an hour or so.) This draws the excess water to the surface, where you can blot it off. It's kind of cool, looks like the eggplant is sweating. You can rinse and dry the eggplant after that to get rid of the excess salt. You probably don't need to bother if there's only a little eggplant along with a bunch of other things, but if eggplant is a major ingredient it does seem to help keep the dish from being watery.
Audrey