From what I've read, applesauce can replace eggs when the eggs are being used mostly for their moisture. Applesauce can't provide structure or leavening (the other things eggs do) just wetness.
I use applesauce in place of dairy products and non-dairy products as I can't tolerate either. Since applesauce is thicker than milk, I sometimes add a little water to it so that the batter has the right consistency. But muffin batters, for example, come in a variety of thicknesses, so you have some wiggle room there. Otherwise I usually make it a straight across substitution. 1 cup of milk = 1 cup of applesauce + water, where the water wouldn't be more than 1/4 cup. If you're replacing sour cream or yogurt with applesauce, it wouldn't need to be thinned with water. If you're replacing a LOT of heavy dairy products with applesauce, you might want to add some oil to make up for the fat you've lost.
Applesauce can replace fat in some cases. I've replaced half the oil with applesauce in gingerbread, quick breads, muffins, and a spice cake, and I couldn't tell the difference in taste or texture.
And if you can't do apples, pear puree should work about the same, or a banana if you don't mind things tasting like banana.
Re: The prospect of cranberry-orange muffins
I use applesauce in place of dairy products and non-dairy products as I can't tolerate either. Since applesauce is thicker than milk, I sometimes add a little water to it so that the batter has the right consistency. But muffin batters, for example, come in a variety of thicknesses, so you have some wiggle room there. Otherwise I usually make it a straight across substitution. 1 cup of milk = 1 cup of applesauce + water, where the water wouldn't be more than 1/4 cup. If you're replacing sour cream or yogurt with applesauce, it wouldn't need to be thinned with water. If you're replacing a LOT of heavy dairy products with applesauce, you might want to add some oil to make up for the fat you've lost.
Applesauce can replace fat in some cases. I've replaced half the oil with applesauce in gingerbread, quick breads, muffins, and a spice cake, and I couldn't tell the difference in taste or texture.
And if you can't do apples, pear puree should work about the same, or a banana if you don't mind things tasting like banana.