Baking zucchini bread without egg yolks
Genetics is actually the greatest affect on cholesterol.  Consumption of dietary cholesterol has very little correlation to serum cholesterol in your body and the amount of cholesterol you're going to save by leaving out the egg yolks vs. the impact it has on the resulting bread will hardly be worth it.

However, if you wish to proceed then I'd recommedn simply adding about a tablespoon of vegetable oil in replacement of the egg yolks that are being omitted.

The fat in the egg yolks is going to assist in minimizing gluten development.  Since you've omitted the fat there is probably a bit more gluten formation taking place and thus the fact that it doesn't rise as well.  Egg whites have a drying effect on baked goods so that is why the bread is drier than before.