Let $A$ be the free Boolean algebra on $\omega$ free generators. Then $A$ is isomorphic to the field of clopen subsets of the Cantor space $2^\omega$, which is the Stone space of $A$.
Let $B$ be the free (Boolean) $\sigma$-algebra on $\omega$ free generators this time. Then, I think, $B$ is $\sigma$-isomorphic to the $\sigma$-field generated by the clopen subsets (or Baire subsets) of the Cantor space $2^\omega$, which is not the Stone space of $B$ since it is the Stone space of $A$.
Let $C$ be now the free (Boolean) $\sigma$-algebra on $\omega_1$ free generators. Is $C$ $\sigma$-isomorphic the $\sigma$-field generated by the clopen subsets (or Baire subsets) of the Cantor space $2^{\omega_1}$? Is $2^{\omega_1}$ the Stone space of $C$?
If $D$ is a free Boolean algebra on $\kappa$ generators, its Stone space is indeed $\{0,1\}^\kappa$: an ultrafilter is determined by which generators are in it, so by a function $f:\kappa \to \{0,1\}$, i.e. a member of this Cantor cube of weight $\kappa$.
So $\{0,1\}^{\omega_1}$ cannot be the Stone space of your $C$. You will want the Loomis-Sikorski theorem for the $\sigma$-algebra case, I suppose.