How do shiny eggs work?

[The8BitzySpider]

Is the shininess of a Pokémon hatched from an egg determined when you collect the egg or when you hatch it? I've heard that you can tell if a Pokémon will be shiny while it is an egg, is that true?

[Lucifer Morningstar]

If Reborn didn't change it from canon, all egg stats; species in some cases, gender, ability and shiny status are determined when the egg is collected. 
As for being able to tell the shiny status of an unhatched egg, I've never heard of that personally. 

[Dreamy]

Yes all the properties of the pokemon are determined when the egg is created, they basically just have an extra flag that marks them as an egg.
Determining if an egg is shiny before it hatched could only be done with some kind of mod or debug access, there's no visible signs of it in the regular game before it hatches.