If I have a white leghorn rooster and an EE hen will my eggs be white or colored ?
If the EE lays colored eggs (blue or green), I would expect either half or all of the daughters to lay blue eggs (or maybe green ones.) Either half or all of the sons should carry the genes for that egg color, which they could pass on to their own chicks.
Whether you get colored eggs from half the daughters, or all of the daughters, depends on the genes of the EE hen.
If the EE lays lays eggs that are not blue or green, and she's mated to a Leghorn rooster, the offspring will not produce any blue or green eggs either. In that case, the daughters will probably produce eggs that are white or a light shade of brown (cream, tan, etc.)
(Just in case you did not know: the EE hen will keep laying the same color eggs, no matter what rooster she mates with. The rooster affects what color eggs her daughters can lay, but has no effect on her own egg color.)
Also being as the leghorns are bred for white eggs and not meat will the chics be dual purpose as the EE are ?
Any chicken can be eaten, even Leghorns.
But I would not expect the offspring to be very good for meat. They will probably have a layer-type body shape and size.