You should get sexlinked chicks, becaues the mother is barred and the father is not-barred.
The sons should be barred (like their mother is), and the daughters should be not-barred (like their father is.)
They will probably be mostly black, with some white or gold feathers showing here and there, and of course the white barring on the males. The gold and the dotted pattern of the Mille Fleur are caused by a combination of genes that are mostly recessive to the all-black base color of the Barred Cochin.
Genetic explanation of how the sexlinkage works:
Barring is on the Z chromosome, and is a dominant gene.
A rooster has ZZ, and passes one Z to each chick. The not-barred rooster is passing not-barred to every chick he sires.
A hen has ZW, and she passes Z to her sons and W to her daughters. So in this case her sons get barring (on the Z chromosome) and her daughters do not (W chromosome does not have any genes that affect barring or not-barring.)
Thus each daughter has Z (not barred) from the father and W from the mother. The daughters are not barred.
Each son has Z (not barred) from the father and Z (barred) from the mother, and because barring is dominant they do show white barring.
