If I bred her with a brown egg layer what would happen?
I'm not 100% sure of her genetic makeup. The blue eggshell gene "O" is dominant over the white eggshell gene "o", so all she needs is one copy of the blue egg gene to lay green eggs. The green comes from the brown overcoating. If you mate her with a brown egg laying breed, one of two things will happen.
If she is pure for blue, which means she has two copies of the gene,"OO", her offspring will get one blue eggshell gene from her and one white from her father., so all offspring will have Oo and the females will lay green eggs.
If she is split (which I would expect) she is Oo and the brown egg rooster is oo. So half her female offspring would have Oo and lay green eggs and half would have oo and lay brown eggs. Of course, this half is just the odds. There is no guarantee that exactly half would lay green eggs. It depends on which gene O or o the offspring randomly gets.
If I cross her with a EE rooster the odds go higher but the laying probably decreases right?
Again, it depends on the EE rooster. Since he does not lay eggs, you don't know for sure if he has any blue egg genes or not. There are three possibilities. He could be oo, Oo, or OO. If he is oo, it is the same as the brown egg rooster above. If he is Oo, the odds go way up, either 75% or 100%, depending on the makeup of the hen. If he is OO, the odds are 100%, regardless of the makeup of the hen.
Now, the egg laying possibilities. There are a lot of myths on this forum about tendencies of EE's. EE's are mutts, not a breed. They do not have tendencies. They are simply chickens that have the blue egg gene, or at least should have it. They can be big, small, any color or pattern, have any leg color or even any comb type. They may lay real well or lay pretty poorly. The eggs may be large or small. It just depends on what chickens they have in their background. Many people think they have to look like Ameraucanas, but they don't. Ameraucanas were developed from EE's, not EE's developed from Ameraucanas. There is an interesting write-up on an Ameraucana Breeders Club website about that.
Whether or not the offspring of mating your hen to an EE results in better, the same, or worse laying depends on the laying tendencies of the parents of the EE rooster. Until a recent dog attack, my best layer was a solid red EE with yellow legs and a wonky sort of pea comb that laid large green eggs practically every day.