Yes. I don't know what are the official scientific names for barring and blue but you can use any letter that will help you remember in private working of Punnet square.. like I made up B=barred and S= "splash" to make it more obvious which gene is being figured out in Punnett. Splash just because the pure form of blue is splash..
He's probably Bb for barring, which would make him BbSs
For the hens, they would be b-Ss for the blues and b-ss for the blacks. Remember that barring is sex linked so hens can only have one copy of it at most. So hens can only be either b- or B- with the - representing the "Y" gene.
For putting on a Punnet square, you will need to do a bigger square with 4 rows instead of the two for a one gene square.
Down the rooster side of the square put: BS, Bs, bS and bs
For the blue hen row, put bS, bs, -S and -s
Black hen row, bs, bs, -s and -s
All boxes that get the - in them will be the female chicks.
Mind you, this deals directly only with the sex linked barring and the blue genes. It does not deal with the lemon blue phenotype nor the solid black/blue hens as we don't know if the rooster & hens are birchens(with the hens dark) or they are all blacks but the rooster is 'leaky'.. or he's a birchen and the hens are blacks. Besides trying to do Punnett with more genes is possible but the square gets BIG lol..
My guess though is that the chicks probably will come out a mix of possibly some solid colored with a bunch of them showing some color on hackles/saddles or breasts.