The answer is yes: there are $2^{\aleph_0}$ countable Boolean algebras up to isomorphism, or equivalently $2^{\aleph_0}$ homeomorphism class of metrizable totally disconnected compact Hausdorff spaces. This is the main result of:

<i> Reichbach, M.
The power of topological types of some classes of 0-dimensional sets.
Proc. Amer. Math. Soc. 13 1962 17-23</i> ([Open link][1]).

It precisely consists of showing that there are $c=2^{\aleph_0}$ closed subsets in a Cantor space, modulo global homeomorphism. Adding a discrete countable subset accumulating onto the given closed subset yields the desired family of continuum many non-homeomorphic metrizable Stone [=totally disconnected compact Hausdorff] spaces.

(Note that it also directly implies that there are $\ge c$ isomorphism types of Boolean subalgebras in $2^\omega$. At the topological level, classifying Boolean algebras embedding into $2^{\aleph_0}$ is the same as classifying [nonempty] separable Stone spaces. For Stone spaces, the class of metrizable spaces is properly contained in the class of separable ones. By Reichbach's 1962 result the former (modulo homeomorphism) has cardinal $c$ while by Freniche's 1984 result given by Juan, the latter has cardinal $2^c$.) 

  [1]: https://www.ams.org/journals/proc/1962-013-01/S0002-9939-1962-0133103-6/S0002-9939-1962-0133103-6.pdf