There do not seem to be any genes that affect both earlobe color and egg color. Or pairs of linked genes that affect the two traits.Is this a myth, or do lobes and eggs actually affect each other, color-wise?
But among pure breeds, most breeds that have white earlobes do lay white eggs. And most breeds that have red earlobes do lay red eggs. There are exceptions both directions (commonly listed exceptions: Dorkings with red earlobes and white eggs, Penedesencas with white earlobes and brown eggs.)
Among mixed breed chickens, if both parent breeds have the same earlobe color and lay the same color eggs, chicks are likely to have those traits too. (So a Rock/Wyandotte cross would probably have red earlobes and lay brown eggs, like both parent breeds.)
If there is a mix between breeds with different earlobe colors, the chicks can have red or white or two-colored earlobes. And if there is a mix between breeds with brown and white eggs, chicks can lay eggs of in-between colors. With a few generations of mixing, you can have chickens with any color earlobes laying any shade of white, cream, tan, or brown eggs.
(I'm agreeing with what several other people have said, just giving the explanation in slightly different words.)