The following result seems to be true (I can prove it, only quite indirectly):
Let $X$ be a Stone space (i.e. a compact totally disconnected Hausdorff space) and $\sigma : X \to X$ be a continuous map with $\sigma^2=\mathrm{id}$ and $\{x : \sigma(x)=x\}=\emptyset$ (i.e. a fixed-point-free $C_2$-action on $X$). Then there is some open subset $U \subseteq X$ such that $X = U \sqcup \sigma U$.
Question 1. Is this well-known? Is there a reference for this result?
Question 2. There are counterexamples when $\sigma$ is allowed to have fixed points (Arens-Kaplansky, Topological Representation of Algebras, Section 8). Is there also a counterexample if $X$ is the Cantor space $\{0,1\}^{\mathbb{N}}$?