Store-bought brownie mixes have evolved with busy homemakers in mind. The mixes rely more on a quick stir-up and baking time than they do on preparation -- usually listing only a few additional ingredients, such as oil and water. For the brownie mixes that require eggs, going without eggs may be possible but not always palatable.
Eggs are more than just a poultry product in brownies -- they act both as congealing and moistening agents. The protein in eggs acts almost as a glue, bringing the brownies together; without the egg, the brownies will crumble. Also, eggs help to retain moisture throughout the baking process, so making brownie mixes without them will make the brownies hard and dry.
If you are caught without eggs, you do have a few suitable substitutes. Both applesauce and mashed bananas can give brownies form and moisture, though they might alter the taste a bit. Substitute a banana or 1/4 cup applesauce for each egg. Another good substitute is ground flax seed; mix 1 part ground flax seed with 3 parts hot water, stir, and let sit for about 10 minutes before adding it to the mix.