Can I replace eggs, oil and water with apple sauce or just eggs?
You can replace any of the three with applesauce, or all of them - with the caveat that it will change the taste and texture of the final product, with more difference from more substitution.  You do not have to substitute all of them if you choose to substitute one of them - the substitutions would be independent of each other (other ingredient amounts stay the same), although the results will stack together.

Substituting eggs with applesauce is fairly common - the pectin replaces the binding action of the egg.  One egg can be substituted with a quarter cup of applesauce.  The effects on the recipe are usually minimal if only one egg is substituted in this fashion, the more eggs in the recipe that are replaced, the bigger the change (tending towards denser since egg lifts and fruit sauce weighs down).

Substituting applesauce for oil is apparently trickier, since they both inhibit gluten but oils additionally act as binding agents in the dough, while applesauce is largely watery and does not bind the dough.  It is usually recommended to substitute out only part of the oil - and not more than a quarter cup total (in a one-to-one ratio) per recipe unless tested.

Substituting applesauce for water is more easily done since the purpose of the water is only to provide moisture, which applesauce has.  It can be substituted one-to-one (though as it is thicker, more liquid might be needed depending on how wet the dough needs to be).  Again you might end up with a slightly denser product, since you are essentially adding solids (more mass) to the recipe and fruit tends to weigh the bread down.

Overall, if you replace all three with applesauce your final product will be heavily apple-flavored, a great deal sweeter, and probably very dense and crumbly.  It will not have the lift from the egg, or the binding from the oil.  It will dry faster, and will suffer more if over-baked.

In the end, I do not recommend replacing all three ingredients in the same recipe.  Replacing one will probably give you the best result, or the result most like the original recipe.  

If you really want a recipe with all-apple-everything, you should probably look up a applesauce-bread recipe to begin with, or perhaps look at some of the recipes for banana bread that use a very high ratio of fruit to bread ingredients.