Substitution of applesauce for eggs in a boxed brownie mix was a failure. Did I do something wrong?
Egg yokes contain molecules called lecithins that act as an emulsifier (something that helps oils to mix with water). I'm guessing the brownie mix calls for canola oil to be added, which is what is causing it to come out oily. You can address this in a couple ways.


Substitute the oil with butter. Butter is already an emulsion itself, and so will tend to separate less from the liquid ingredients. This will make the brownies thicker when cooled, so the mouthfeel will be less moist.
Substitute 1 Tbsp soy lecithin (available from Amazon) per egg yolk. This is the most direct way to compensate.


Egg whites also have useful material properties for baking. They act as a leavening agent and also help provide a set texture when the protein denatures at high temperatures. These are harder to compensate for but also less critical than emulsification. Try these next ideas if you are still unsatisfied with the texture.


Add baking powder if you want the brownies fluffier.
I've never tried this personally, but apparently substituting 1/4 cup pureed silken tofu per egg will help your batter set as though it had egg whites. It makes sense since both are basically networks of hydrophilic proteins.