So the white rooster is the offspring of the cuckoo silkie and the white jersey giant, right? In that case, he should have inherited one gene for barring (assuming the silkie has two copies of the barring gene). So under the white, this rooster is barred. Assuming the white in the jersey giant hen is not hiding barring, the white rooster will have one gene for barring. He will have a 50% chance of passing barring on to his offspring, regardless of whether the chicks are male or female.
If jersey giants are recessive white, then the silkie rooster must also be carrying a copy of the gene for recessive white. The white rooster must be carrying two copies of recessive white, since he himself is white. Therefore he will pass a single copy of the recessive white gene on to every one of his chicks, both male and female.
If the female is also a result of the cuckoo silkie rooster and white jersey giant then she has a single copy of barring (females can only ever have a single copy) and a single copy of recessive white. She will pass the barring on only to her male offspring. But she has a 50% chance of passing recessive white on to any of her offspring both male and female.
From this pairing, each chick has a 50% chance of inheriting 2 copies of recessive white (and therefore being white). Each female chick has a 50% chance of inheriting a copy of barring from the white rooster and being barred (assuming they don't inherit 2 copies of recessive white).
Female chicks also have a 50% chance of inheriting no barring gene, which would make them black (possibly with leakage once they start feathering in depending on what else the white is hiding).
Male chicks will inherit one copy of barring from the hen. They have a 50% chance of also inheriting a second barring gene from the rooster. So all of the male chicks from this cross will have at least one barring gene. However, since both parents can pass on the recessive white, they won't necessarily all be barred. Some of the male chicks from this pairing may be white.
-For future breeding from these chicks keep in mind that some of the males may only have one copy of barring and can pass non-barring (solid black) on to their offspring. Some of the males may have two copies of barring and these will pass barring on to every one of their offspring. For the barred chicks, the darker males will be the ones with only one copy and the lighter colored males will have two copies. Any white males from this copy will carry at least one copy, but since white hides barring you won't be able to tell which ones have only one copy and which ones have two copies.
Barring causes white bars on the feathers. It can be present on any color. Traditional barred/cuckoo, however, is on a black base. Black gives the sharpest contrast (barring on some colors, such as blue or brown can be subtle enough as to not be noticeable, especially if the bird only has one copy of the gene for barring).
I don't know enough about melanism inheritance to comment on it(i think its dominant, but i don't work with melanistic breeds so i haven't really looked into it), but I think it's possible the chick is not melanistic, just black. Black birds often have black legs and beaks and it's not uncommon for their combs to be black when they are very small but they turn pink/red as the chick matures.