Without knowing what genes are in the white male it is not really possible to tell whether any of the offspring would be blue. It is possible, but not really usual, for a white bird to carry blue under white.
Since I know absolutely nothing about genetics what I am saying probably means nothing, but you couldn't use a BSL as a "black" bird anyway. Could you? A BSL isn't black, its barred and isn't even a pure barred bird. Seems that is about as far from black as you are going to get?
Woah, edit because I thought he said BSL roo.....but either way, there are no pure black parents in the sex link.
In order to turn plumage blue which would otherwise be black one only needs one blue gene. his can be from a splash, which carries two blue genes, it could also be hidden under a white. A red or buff bird could carry one or two blue genes. Only the parts which would normally be black in such birds would show as blue.
Some offspring of a black sex link could be expected to be black & thus be affected by a blue gene if present.
All barred or cuckoo birds are actually solid black chickens.. It's the barring gene that puts white bars across the feathers.. making them barred/cuckoo.
This is why BSL hens are black- they got the 'black gene' from their Barred Rock mothers. Barring is sex linked so those daughters don't get that gene.. and that is why they are black and not barred.. also is why the sons are black with barring.. they got both from their mothers.
"black and white=blue" often is due to splashes being referred to being 'white' in basic genetic lessons. But in reality these 'white' chickens are Splash.. not pure white as in white leghorns or white rocks. White and splash are completely different genes(but both can be found in the same bird as Krys mentioned. The reason being blue can really help "clean up the white" by getting rid of any miscellanous black flecks).
The other part not mentioned here, it also depends on what kind of white the rooster has. There are two major versions of white- a recessive and a dominant white. Most white leghorns have dominant white.. so if you are using a white leghorn over that BSL hen, then most likely all chicks will be cream with random little black spots and grow up mostly white.. some may show a little brown/red bleeding through on some areas.