This odor is usually associated with harmless hydrogen sulfide gas. It can occur naturally in well water or may be caused by bacteria or chemical reactions in your water heater.
A rotten egg smell sounds like it would be sulfur or gas- I am not sure why your laundry would have this smell, though. Are you sure the smell is directly related to the laundry, or is it perhaps something else?
Usually a rotten egg smell is caused from using well water to run your washer. If this is the problem it would be next to impossible to avoid this smell. The best thing to do would be to go to the closest laundromat and do your laundry there. Few extra dollars but your laundry will smell fresh.