= Solution
A <zero-dimensional space> has a base of <clopen sets>. A <totally disconnected space> has no <connected subset> with more than one point, equivalently all its <connected components> are singletons. We use these definitions in the stated Hausdorff setting.
If a <Hausdorff space> has a clopen base, any two different points can be separated by a <clopen set>. A <connected subset> containing both would be split into two nonempty disjoint relatively open sets. Thus \b[zero-dimensional implies totally disconnected] here.
\b[The compact converse, including the component lemma.] For a point $x$ of a compact Hausdorff $K$, put
$$
Q_x=\bigcap\{N:x\in N,\ N\text{ clopen in }K\}.
$$
This is its <quasicomponent>, a closed compact set. We prove the <Šura-Bura component lemma> that $Q_x$ equals the <connected component> of $x$.
Suppose $Q_x=A\cup B$ were a separation into nonempty disjoint relatively closed sets, with $x\in A$. Both are compact, hence closed in $K$. Normality of <compact Hausdorff spaces> gives disjoint <open neighborhoods> $U\supseteq A$, $V\supseteq B$. The compact set $H=K\setminus(U\cup V)$ misses $Q_x$. For each $y\in H$, choose a clopen neighborhood $N_y$ of $x$ missing $y$; these exist by the definition of $Q_x$. The open complements cover $H$, so finitely many suffice. Their intersection $N$ is clopen, contains $Q_x$, and lies in $U\cup V$. If $H$ is empty take $N=K$.
Now $N\cap U$ is clopen in $K$: it is open, and its complement is $(K\setminus N)\cup(N\cap V)$, also open. It contains $x$ and misses $B$, contradicting that $B\subseteq Q_x$ lies in every clopen neighborhood of $x$. Hence $Q_x$ is connected. Conversely every <connected subset> containing $x$ lies in every clopen neighborhood of $x$, so the <connected component> is contained in $Q_x$. Equality follows.
If $K$ is <totally disconnected>, then $Q_x=\{x\}$. Given an <open neighborhood> $O$ of $x$, for each $y\in K\setminus O$ choose a clopen neighborhood of $x$ missing $y$. Compactness again gives a finite intersection $N$ with $x\in N\subseteq O$. These $N$ form a clopen base. Therefore
$$
\boxed{K\text{ compact Hausdorff}:\qquad K\text{ zero-dimensional}\iff K\text{ totally disconnected}.}
$$
\b[The Stone-Čech characterization.] The stated clopen separation property is <strong zero-dimensionality> in this normal Hausdorff setting. A normal <T1 space> is a <Tychonoff space>, so its <Stone-Čech compactification> exists. We use its extension property for maps into <compact Hausdorff spaces> and the density of $X$.
First assume $\beta X$ is a <zero-dimensional space>, and let $C,D$ be disjoint <closed subsets> of $X$. <Urysohn's lemma> gives $f:X\to[0,1]$ equal to zero on $C$ and one on $D$. Its continuous extension to $\beta X$ puts $\overline C^{\,\beta X}$ and $\overline D^{\,\beta X}$ in disjoint zero and one level sets. Using a clopen base, cover the first compact set by finitely many <clopen sets> missing the second; their union $H$ is clopen and separates them. Thus $U=H\cap X$ and $V=X\setminus U$ are disjoint <clopen sets> containing $C,D$.
Conversely, assume the stated clopen separation property on $X$. Let $A,B$ be disjoint <closed subsets> of $\beta X$. Normality supplies a <continuous function> $f:\beta X\to[0,1]$ with $f|_A=0$, $f|_B=1$. The sets
$$
C=X\cap f^{-1}([0,1/3]),\qquad D=X\cap f^{-1}([2/3,1])
$$
are disjoint and closed in $X$. Select clopen $U\subseteq X$ containing $C$ and missing $D$; the given disjoint pair of clopen neighborhoods supplies such a $U$. Its <characteristic function>, one on $U$ and zero outside, extends from $X$ to a <continuous map> $\beta X\to\{0,1\}$. Let $H$ be its one level set, a <clopen set>.
Every $a\in A$ is in $\overline C^{\,\beta X}$: any neighborhood of $a$, intersected with $f^{-1}([0,1/3))$, is an <open neighborhood> meeting the dense set $X$. Likewise $B\subseteq\overline D^{\,\beta X}$. Since $C\subseteq H$ and $D\subseteq\beta X\setminus H$, it follows that $A\subseteq H$ and $B\cap H=\varnothing$.
Apply this to $A=\{z\}$ and $B=\beta X\setminus O$, for any point $z$ and <open neighborhood> $O$. It yields a clopen $H$ with $z\in H\subseteq O$, proving that $\beta X$ has a clopen base. Hence the <strong zero-dimensionality and Stone-Čech compactification> criterion is
$$
\boxed{X\text{ has the stated clopen separation property}\iff\beta X\text{ is zero-dimensional}.}
$$
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