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A partial answer to the question about the Gleason cover of the unit interval: it can be found everywhere in every compact extremally disconnected space. The point is: in a compact extremally disconnected space the closure of every countably infinite relatively discrete subset is (homeomorphic to) $\beta\mathbb{N}$ and $\beta\mathbb{N}$ contains homeomorphic copies of all compact extremally disconnected spaces of weight $\mathfrak{c}$ (or less). Sketch of proof of the embedding result: the Cantor cube $2^\mathfrak{c}$ is separable, hence there is a continuous surjection $f$ of $\beta\mathbb{N}$ onto the cube. Take a copy in $2^\mathfrak{c}$ of the e.d. space $X$ that you want to embed and apply Zorn's Lemma to get a closed subset $K$ of $\beta\mathbb{N}$ such that the restriction $f\mathbin\upharpoonright K$ maps $K$ irreducibly onto $X$. Because $X$ is e.d. this irredicible map is a homeomorphism. Even nicer embeddings are constructed in this paper