The odds of you getting colored eggs really goes up if some of those chicks are pullets.
The color of eggs you might get really depends on the genetics of the rooster. With an EE rooster, you often don't know what his genetics are.
Chickens have a pair of genes that determines the base color of the eggs. That base color will be either blue or white. Since blue is dominant, if just one of the two genes in that pair is blue, the base color will be blue. You can look at the inside of the shell to see the base color. Since your rooster does not lay eggs, you can't tell by looking at him.
There are a lot of different genes that determine brown. Which genes the chicken has will determine how much, if any brown. The way it works:
Base white + no brown = white
Base white + brown = brown
Base blue + no brown = blue
Base blue + brown = green
Since all your hens lay brown eggs, the eggs will be either brown or green.
If the EE rooster has two blue genes, all his offspring will inherit one blue gene and all the pullets will lay green eggs.
If the EE rooster has no blue genes but only white, all the pullets will lay brown eggs.
If the EE rooster has one blue gene and one white gene, Some of his offspring will inherit a blue gene and some will inherit a white gene. So his offspring will either lay brown or green eggs with your hens.
So depending on the genetics of your EE rooster, the odds of getting green eggs are 100%, 0% or 50-50 with each pullet.