I have a couple of followup questions.
Re blue eggs, ... are you saying an Easter Egger cannot carry Blue/Blue?
Yes - clarificiation is needed there - I used the term Easter Egger to mean a chick or cross or descendent from an Araucana (or Americauna dependent on where you are) that has been bred to another non Araucana breed. In my head an Easter Egger describes a Non-Pure breed bird - but of course the "reason" people breed them is to get the blue eggs so that is why I referred to it in my example - since I wanted to talk about blue egg genes.
To avoid confusion - I might continue with the terminologies "Blue Egg Layer" or "Olive Egg Layer" for clarity
To answer your question above:
Let's take a blue egg laying hen with double blue egg genes (Blue / Blue) and breed her with a rooster with white egg genes ( - / - ),
Mom will contribute one blue gene to 100% of the chicks
Dad will contribute the "-" gene to the chicks.
So 100% of your chicks will be (Blue/-) Blue egg layers.
If your mom hen above was an Araucana (Americauna depending on location) and your rooster was something else - We would commonly call their chicks "Easter Eggers".
Let's take one of their rooster babies and see what happens if we breed him to a blue egg laying hen of another breed (non Araucana) (so moving from the F1 generation to the F2 generation)
Rooster baby from above that contributes (Blue / -)
Blue egg laying hen contributes (Blue / Blue)
Half of the chicks will be (Blue/Blue) and the other half (Blue/-)
They are all still blue egg layers so yes it is possible to achieve second generation "mixed breed" babies that are blue egg layers with Blue/Blue genetics. You just have to select the right birds that are known to carry the blue egg genes.
When you do these mixed breedings and combine blue egg genetics with brown egg genetics - that's where you get babies that produce "Olive Eggs" called "Olive Eggers".
My question about speckled eggs is shorter. Are you saying in order to have speckles, an egg needs speckled genes from Both parents?
I'm actually not that familiar with the "speckled" genetics - I do not have it at all in my flocks. That's a question best posed to the BYC community that have speckled egg layers

But I did want to mention it so that you had an example of another "effect" that can be seen on the eggs. Some people deliberately breed for the speckles - they're kinda cool!