Not for the statistics.
"Shiny Pokémon are always going to end up with the same odds of good stats as normal Pokémon. They used to have set IV's in the Metal generation, but not in this one."
--Serebii
They aren't going to have better statistics. If you want a shiny Pokemon because it looks cool, that's fine, but don't do it for the stats, they're the same for all Pokemon except in Generation 2. And they don't have better EVs either, since EVs can only be obtained through training, and no bred Pokemon Starts out with EVs. They also have no greater EV capacity.
Furthermore, Pokerus is the same when spread to all Pokemon. Pokerus doesn't make your Pokemon stronger in the end, it just increases the speed your Pokemon get EVs. It doubles the EVs you get from defeating a Pokemon, but the limit for how many EVs you can obtain is still the same.
If you ARE breeding for Pokemon with better stats, I would recommend finding a Pokemon in your starter's egg group of the opposite gender with the correct nature, and giving it an everstone when breeding it, passing down the nature. If you can find a Pokemon with such a nature and maximum IVs in one of your Pokemon's Crucial statistics, then you can give it the corresponding Power Item, passing down the IVs as well.
