Can I use egg white instead of whole egg in brownie mix? I plan on using apple sauce instead of oil too. Will it all work?

The changes you're thinking about will work, but the question is how well you'll like the results (so subjective).  Using egg white will change the texture of the brownie in a noticeable way - the texture will be a little more adhesive or sticky.  If you're OK with that change, increase eggs by 1 - ex.:  if the directions call for 2 eggs, use 3 egg whites for the euivalent moisture.  Replacing the oil with applesauce will now remove all the added fat.  Fat makes baked goods tender and tasty, also being a flavor carrier.  I usually recommend replacing half the fat with applesauce for the first try.  Then, if you're happy with the flavor and texture, try all applesauce the next time.  When I remove all added fat, I boost the flavor by adding a teaspoon of vanilla.  For more lighter baking ideas, check out Healthy Baking Tips and Low-Fat Substitutions.