No, at least not in the same game.
When you recieve an egg, it's IVs, Nature, and gender are determined and set in stone. You definitely can't change that. Shininess is slightly different. While soft resetting the one (or 5) eggs on the one game will never roll a shiny, you could trade your eggs to another game and have another go at hoping it's shiny.
Source
Once you receive an egg, every stat and value about the Pokemon inside has already been determined, however some values such as shininess can be altered by trading to a different game. In Gen 8+, this extends even further, in that every time the egg is traded, it's PID is reset, so you can trade back and forth between the same two games and still get a new shiny chance every time.
Hope I could help!