If you save before recieving the egg, you can soft-reset until it's shiny. This works in every gen where eggs are available.
However, once the egg is in your party, everything is pre-determined. Nothing about the Pokemon is determined prior to accepting the egg.
Source: Crystal's mystery egg at the daycare; I used to soft-reset for Pichu, learned that if I saved before accepting the egg the result would be different, but if I saved with the egg in my party it'll be the same. Also in-general experience with eggs and shiny hunting. Lots of trial and error.