Egg is a slur used to describe Whites in a derogatory way. White males that try to really get into East Asian culture and date Asian women. White on the outside, yellow on the inside (opposite of banana).
The word Egg is a racial slur that has been used historically to degrade and dehumanize whites. The word is still used by some people for this purpose, and it is seen as offensive and hurtful to whites. If you are unsure about whether or not it is appropriate to use the word Egg, it is best to err on the side of caution and avoid using it in any situation.