For example, F1 should lay all green eggs due to the cross
F2 could lay any color but who knows what color the cockerals "lay" so they are your mystery element, if you cull all hens that lay different color eggs, then breed them back to your green layer rooster, then all of their kids in the F3 generation SHOULD have just the green egg genes because you know their mothers only have green genes (they lay green eggs) and you know their father is also a green egger due to his breed.
After that F3 generation you should be able to breed them between themselves and back to the green eggers from the starting generation to get the traits you need as long as you don't outcross again to another color of egg layer the green eggs should be more or less fixed in your "breed" you are starting.
Um, that doesn't quite work.
F2 hens that lay green eggs could have OO (pure for blue egg gene) or Oo (split for blue egg gene.) So some will breed true for the blue egg gene, but some are just as mixed as the F1 hens were.
Crossing those F2 hens back to OO rooster (pure for blue egg gene) will give some that are pure for the blue egg gene, but some could still produce Oo chicks (impure for blue egg gene.)
You could test individual F2 chickens, of either gender, to see whether they are pure for the blue egg gene, although it's a nuisance because it takes a lot of time & space. Cross each one to a bird that is pure for not-blue, hatch a bunch of chicks, raise pullets to laying age, and see if any lay not-blue eggs. If you have get at least half a dozen daughters, and all of them lay blue or green, then it's a fair guess that their one parent was pure for the blue egg gene. (At which point you can use the parent in your breeding program, but you may have no use for all those pullets that were produced as part of the test--you know they have at least one not-blue gene because that was required to make the test work.)