In that case, yes the mother must be a Barred Rock, and the chick must be female.
Whether the chick will lay brown eggs or green, I can't say for sure.
Since the mother laid brown eggs, the chick inherited a not-blue eggshell gene from her. But the father might or might not have any blue egg...
Barring is a dominant gene on the Z sex chromosome.
A rooster has chromosomes ZZ, a hen has ZW.
Your rooster does not have barring, so he gives every chick a Z chromosome with no barring.
The barred hen would give her Z chromosome (with barring) to her sons, and they will show barring because...