How to figure the calories in a boxed cake mix if applesauce is used instead of egg and oil?
The nice thing about calories is that they are very constant, mixing or combining things does not change the sum of calories.

So in short, calculate the calories you add (the applesauce) and the calories you leave out (eggs, oil). Add and subtract these from what the box says and you're done. 

If the box gives calories per unit or serving, you'd first have to calculate the calories for the whole box, then do the adding / substracting step, then, if wanted, divide the results into servings again. (The Rule of Three comes handy in these cases.)