While breast feeding it is important to consider the baby's health. Hair color and other chemicals are not a good idea. Highlights or lowlights may be a good replacement for color that is applied directly on the scalp. This means, as long as the color does not touch the skin it may be OK. Consider your doctors advice.
Hair may fall out during breast feeding this has to do with your hormones and not because of breast feeding. Remember it's temporary and your hair should grow back within six months to a year.