The Yolk and White are very different and I would expect to see a significant difference in result.
That isn't to say it will be bad, just different.
The Yolk is the source of fat. It is going to impart flavor to the dish and provide a creamier texture.
The White is mostly protein and so will set more, have more of a texture impact.
If it called for several whole eggs and you wanted to take a Yolk or two out for less fat, that would have less of a variance, but switching several Yolks to several Whites should definitely alter the results.
Extra info:
Here is some nutritional information of a Large Whole/White/Yolk.
http://www.incredibleegg.org/health-and-nutrition/egg-nutrients/nutrient-chart
You will see they both carry protein however the Yolk has almost twice as much fat as protein while the White has less than 2% of fat as it does protein.
This makes their behaviors very different.
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.
Best Answer
This is not a real answer, but rather some info to hopefully, help:
Egg whites are 90% water, when they harden, the protein unravels and creates a grid network that does not allow the molecules to slip past each other. If the protein level drops, it might make it harder to set and stay runny.
Egg white proteins go through a phase change (become hard) at a lower temperature than the Yolk by a few degrees, that's how you can have soft-boiled eggs.
If you crack a fresh egg and it sprawls on the dish (as opposed to hold tall), you're looking at an egg from a chicken that didn't eat many insects/larvae. ;) This may be expected as at this time of the year (northern hemisphere) insects aren't very active.
There is a good section in Michael Pollan's Omnivore's Dilemma that gets into more detail. And perhaps better information regarding the off-season feed.