Comment by Gro-Tsen on Stone-Čech compactification of a Boolean subalgebra of $\{0,1\}^S$

Thanks! This was exactly the explanation I needed. Both papers are quite interesting, and, crucially, I had missed the fact that when $B$ is countable it can be partitioned into countably many nonempty clopen sets. I just wrote a streamlined proof on MSE that $B$ countable implies that $\beta B$ cannot equal $\{0,1\}^S$.