Another reference for "Answer = Yes":
Chapter 12 of the Handbook of Boolean algebras has the title "The number of Boolean Algebras".
Don Monk, the author of this chapter, writes: "For almost all classes K of BAs which have been an object of intensive study, there are exactly $2^\kappa$ isomorphism types of members of K of each infinite power $\kappa$."
In particular, this is true for K=interval algebras. There are $2^\kappa$ many linear orders $L$ of cardinality $\kappa$ such that the corresponding interval algebras $Int(L)$ are pairwise non-isomorphic. (The elements of $Int(L)$ are the finite unions of intervals of $L$.)