Whole Egg, Egg White or Egg Yolk for binding Hamburgers
Generally you'd use the whole egg. The egg white provides most of the structural integrity, but the yolk does a little too. Also, the yolk helps with what might otherwise be a rubbery texture. Letting the patties sit in the refrigerator for half an hour or so will also help them stay together.

I suspect that the answer you linked, talking about how the yolk is the "only binder", is mistaking emulsification for binding. Ground meat, veggies, and egg whites are all water-based mixtures; they don't need lecithin to play nice with each other.

Incidentally, if you like zucchini and carrots and onions inside the burger, you'll like them even more on top of the burger. Take all that stuff you were going to mix in (other than the egg), slice the veggies rather than mince them, add a bit of olive oil and more salt than you think you should, and saute the crap out of them on high heat for a few minutes. Really char it up. Vegetables in a burger don't want to be steamed in a jacket of meat; they want to do their own flavor work.