Nah, don't worry. Changing the Pokemon in the Daycare will have no effect on your progress. The only thing that would have an impact would be resetting the game, as it would generate a new seed. (Or possibly turning on/off the C-Gear). And even then, you'll just have a different frame you'll need to hit.
When you turn on the game it does not specify that you will need to hatch X amount of eggs to get a Shiny. It's more like every egg you hatch has X chance of being Shiny.
Good luck!
Thanks a bunch! I just replaced my ditto with a female modest omanyte holding an everstone because apparently modest is a good nature for an omastar. I'll hopefully find my shiny faster and I actually want to find it now because I realized that Omastar is actually pretty decent. I'm at about 300 eggs now from my omastar and ditto and hopefully I'll find my shiny soon.