Replacing egg whites with whole eggs when baking cookies

Egg yolks and egg whites have very different roles in baking. You can almost never replace them. And in the cases in which you can, you will end up with a different texture.

In this case, it won't be sure disaster to replace, but I would be very reluctant to do it. Flourless recipes are finicky. Flour holds stuff together. Nut flours don't hold anything together, they need a binding agent. Egg whites are a good binding agent. Egg yolks are not only not good as a binding agent (except in certain circumstances, for example in custards - but you don't have this here), they are even a lubricant and as such will interfere with the binding done by the egg whites.

If you substitute here, you will not only change the taste, you risk your cookies crumbling apart in your hands.

In general, don't replace stuff in baking recipes. They are hard to design right. Even if you have some issues (e.g. allergies), it is easier to find an existing recipe without the offending ingredient than to try to tweak an existing one to work with different ingredients. Eggs are especially hard to substitute. Such a reason like "not wanting to waste" loses its sense if what you end up with is bad cookies, which are a much worse waste of products. And besides, you can do many more interesting things with a yolk than just scrambled eggs. In fact, I am frequently throwing out whites because I have found no use for them after doing something with the yolks.

If you still want to go ahead and try if the cookies work for you, try using 3 whole eggs. 2 eggs are nowhere near enough to substitute for 4 egg whites. Not only are the yolks smaller than the whites volume-wise, they will also not deliver you the binding proteins you need so badly.

Whenever you consider using whole eggs instead of just whites, first consider the reason it's an egg white-only recipe: is it purely for nutritional (fat/calories) reasons? Or, is it for volume? In this case, it seems to me that it is the latter, therefore, I would separate the eggs, mix in the yolks with the other ingredients and then whip the eggs separately, to a soft meringue stage and then fold them into the chocolate mixture (after the yolks are mixed in).

Egg yolks are about 30% of the egg, so if you added whole eggs you'd need to add 3 instead of 4 to get the same approximate weight of egg. As for what the results would be I doubt you'd be able to taste much difference. Egg yolks are much richer than white, however that richness of flavor would be overshadowed by the coconut, chocolate, and sugar.

As the eggs are the main binding agent changing the consistency of the eggs will change the consistency of the result, but again I don't think you'll see much difference. The egg whites are not whipped, so I wouldn't see why this wouldn't work.

In a nutrition class, I was taught to use two eggwhites instead of a whole egg, to avoid cholesterol. You should not have to modify the recipe any further.