In simplistic terms, a rooster gives a gene from every gene pair he has to all his offspring, boys and girls. A hen gives a copy of one of the genes at every location on the chromosome to her sons, but does not give the sex-linked genes to her daughters. That’s why the hen determines the sex even before the egg is fertilized. That’s the basis for sex linked.
Nothing is ever straightforward with chicken genetics, there are always exceptions to practically anything. But in general, this means the rooster has as much and in some cases more influence than the hen in everything.
Egg shell color is a good example. There are several different genes that determine egg shell color, some dominant and some recessive. If the egg is brown, what shade of brown depends on which genes are present and if they are dominant or recessive. The hen and rooster both contribute to that. But one of those genes is sex linked. If he has it the rooster can give it to his daughters and sons, but if the hen has it she cannot give it to her daughters, only her sons. So if the gene is present the rooster might have more influence on egg shell color than the hen, but if the gene is not present, they contribute equally.
I don’t know of any sex linked gene related to the size of an egg, so the hen and rooster probably both contribute equally to that. But a rooster does not lay eggs. How do you know what he is contributing? If you know what egg he hatched out of and even better what size eggs his parents hatched out of you have some clues as to what he might have, but you don’t know for sure. So the best way to increase the size of the eggs your flock lays is to save your breeding roosters from the ones that hatched from large eggs and pay attention to the size egg the hen lays that you breed him to.
Who controls eye color in a human, the father or the mother? Both, depending on what genetics they contribute and the genetic laws of dominant and recessive. The same is true for egg shell color or egg size, both depending on the genetic laws and what they contribute.