There is some notational subtletly here. Strictly speaking, BB(n) represents the output of the BB function for the input n. In that sense BB(n) is computable.
However, the BB function itself is not computable. It makes sense to ask "for what inputs is the BB function computable?". When people talk about BB(n) not being computable, they are really talking about the computability of the BB function at n, not the computability of the integer that happens to be BB(n).
However, the BB function itself is not computable. It makes sense to ask "for what inputs is the BB function computable?". When people talk about BB(n) not being computable, they are really talking about the computability of the BB function at n, not the computability of the integer that happens to be BB(n).