You cannot use sex-links to make sex links. That only works once, then the genetics are not right to do it again. I don’t know what your EE rooster looks like, but if you cross the offspring you are likely to get several different colored and patterned chicks. It just depends on what genetics the EE rooster has to start with.
Barring is a dominant gene that is also sex linked. The rooster has two copies of this gene, but the hen only has one copy. The rooster will give a copy to both his sons and daughters. The hen will give a copy to her sons but none to her daughters. That’s what makes it sex linked if the hen is barred and the rooster is not barred.
Since it is dominant, if there is just one copy present, the chicken will be barred. That makes it a little harder to get pure barred roosters. Since the hen only has one copy, she is either barred or she is not but the rooster may be pure or he may be slit for this gene. But if a rooster has two copies of the gene, he should be a lighter color than the hens or roosters that are split for the barring gene, so you can usually tell if you have a few to compare.
Using B to represent the dominant “barred” and b to represent the recessive “not barred”, your barred rock hen that is barred will have only one gene. You can represent that by B,-. The EE rooster that is not barred will be b, b. Their female offspring will be, b, - and their male offspring will be B, b. The female offspring will not be barred but the males will be split for barring.
If you cross the offspring, the hen will give a “b” to all her sons and nothing to her daughters. The rooster will give a gene to all his offspring, but it might be a “B” or a “b”. About half the male and half the female offspring will be barred and half will not. So if you again cross this second generation, picking barred hens and barred roosters, you will get some female offspring that are barred or not. The females are easy. You will get some males that are B, B, some that are B, b. Since you used a barred hen, all the roosters will have at least one Barred gene. So out of this generation, you pick only barred hens and the lighter colored roosters and you probably have barring set in your flock.
But I think you want colored eggs too. The “blue egg” gene that gives you the color is not sex linked. Both parents have two copies of that gene. The blue egg gene is also dominant, so if one copy of the blue egg gene is present, then the hen will lay blue or green eggs. Since the barred rock will put a layer of brown over the base blue egg shell, the eggs will be green. Use “O” for the dominant blue egg gene and “o” for the not blue, which defaults to white.
Since an EE is a mix, you’re not sure of the genetics they rooster has. It may have two copies of the blue egg gene, O, O; it may be split for the gene, O, o; or it may not have the blue egg gene at all. The Barred Rock will not have the blue egg gen so she will be o, o. There are too many possible combinations to go into great detail, so I offer two suggestions. Only hatch green eggs if you want to set the blue egg gene in yours. And always select for the pea comb. The blue egg gene is not really tied to the pea comb, but it will follow it about 97% of the time so if you get roosters with a pea comb, the odds of it having the blue egg gene are pretty good. Since the rooster does not lay eggs, you won’t know if it has the blue egg gene or not unless you hatch chicks from it.
Another complication is that the pea comb is not always real clear. If the chicken is split for the pea comb gene, which means it has one pea comb gene and one not pea comb gene, you can get a wonky looking comb that is somewhere between a pea comb and a single comb. Or if the rose comb gets mixed in, it looks different. Then there are some modifiers that can make it look different too. But if you can identify the pea comb in the roosters, you have a better chance of getting the blue egg gene.
I know this is long and drawn out. The easiest thing to do is to always breed barred hens (once you get some) and barred roosters to set the barring. Only hatch green eggs to set the colored eggs. It will take a few generations to get the barring and green eggs consistently, but you can do it.
Good luck!!