Given $X$ compact and Hausdorff, $C$ a component of $X$, and $U$ open set containing $C$, show that $\exists V$ clopen such that $C\subset V\subset U$

The answer by user254665 shows where you went astray.

HINT: You know from the first part that $C$ is the quasicomponent of $x$. (There is also a proof here.) Now use the fact that the quasicomponent of a point is the intersection of all clopen sets containing $x$. (If this isn’t your definition of quasicomponent, you’ll have to show that it follows from your definition.) Thus, if $\mathscr{H}$ is the family of all clopen sets containing $C$, we have $C=\bigcap\mathscr{H}$.

For each $H\in\mathscr{H}$ let $\widehat H=X\setminus H$. For each $y\in X\setminus U$ there is an $H_y\in\mathscr{H}$ such that $y\in \widehat{H_y}$. Clearly $\{\widehat{H_y}:y\in X\setminus U\}$ is an open cover of the compact set $X\setminus U$, so ... ? In case you get completely stuck, I’ve completed the answer in the spoiler-protected block below.

So there is a finite $\mathscr{F}\subseteq\mathscr{H}$ such that $\{\widehat H:H\in\mathscr{F}\}$ covers $X\setminus U$. But then $\bigcap\mathscr{F}$ is a clopen nbhd of $C$ contained in $U$.

(The inclusions in the question are non-strict, i.e., what I would write $\subseteq$.)