No, everyone is saying the same thing
a cross breed offspring hen will have an egg dependent on what genes she gets.
She may get just what her father carried (what her aunt's showed as egg color), or what her grandfather carried (the color of the egg her great aunt shows) instead due to...
Right, you got it now!
Now remember, 'painted' eggs (brown specifically) will get lighter as the laying season goes on, but after moult in fall the eggs will be dark again.
yes, every hen's egg color is programed in her when her parents mate - her mate has nothing to do with her egg color, just her children's egg color.
you hen and roo's daughters will have lighter color eggs... your hen and roo's sons will not lay eggs.