Replace eggs in a brownie recipe with 1/4 cup of fruit puree or yogurt per egg in the recipe. These egg substitutes add moisture and bind the ingredients, but do not help with leavening as eggs do. Fruit and yogurt egg replacements also perform the basic duty of oil in brownies, turning this into a somewhat healthier treat.
The oil in brownies helps make them fudgy and tender. If using an egg substitution plus the full amount of oil in the recipe, the resulting brownies are overly soft and difficult to remove from the pan. The oil is usually removed entirely in attempt to make the texture consistent with the standard recipe. While it is possible to still add oil to the recipe, adjust the amount so the combined total of egg substitute and oil is not greater than the amount of oil called for in the recipe.
Applesauce is perhaps the best known and most common egg and oil substitute for brownies, but it is not the only option. Mashed bananas, pumpkin puree, pureed prunes and even mashed black beans can be substituted at a ratio of 1/4 cup puree for each egg. These fruit-based substitutes alter the flavor slightly, adding some of the fruit flavor with the rich chocolate brownie flavor.
Yogurt also works well as a substitute for eggs in brownies. Use plain yogurt so the substitution doesn't change the flavor. When using yogurt made with full milk fat, the lack of oil in a recipe goes unnoticed.
If there simply is not any eggs or oil available to make brownies, use mayonnaise equivalent to the amount of oil in the recipe and reduce the amount of water. Mayonnaise is made with eggs, oil and vinegar, so it is a logical substitute.