I think I can now give a positive result, using some of the extra assumptions ($X$ compact Hausdorff, $Y$ open, $Z$ closed). It will be convenient to use the following:
Lemma
Let $X$ be a Hausdorff space and $C \subset X$ have a compact neighbourhood $K$. Then $C$ is a component of $X$ if and only if $C$ is a component of $K$
Proof of `only if':
If $C$ is not a component of $K$, then $C$ is not connected, or there is a connected subset of $K$ that is a proper superset of $C$. Either way, $C$ is not a component of $X$.
Proof of `if':
Assume $C$ is a component of $K$ and let $B$ be the boundary of $K$ in $X$. Clearly $C$ is connected, so we need to prove that no proper superset of $C$ is connected.
Let us consider $K$ as a subspace. Since $K$ is a compact Hausdorff space, $C$ is a quasicomponent. (for a proof see this answer) Because $C \cap B = \emptyset$, this means that for every $b \in B$ there is a clopen neighbourhood $U_b$ disjoint from $C$. These neighbourhoods form a cover of $B$, that by compactness has a finite subcover. Let $U$ be the union of this subcover. Being a finite union of clopen sets, $U$ is clopen and so is its complement.
Because none of the $U_b$ intersect $C$, and because $B \subset U$, we have $C \subset K \setminus U \subset K \setminus B \subset K$. Since $K$ is closed in $X$ and $K \setminus B$ is open in $X$, $K \setminus U$ is clopen in $X$ too. We may conclude that any connected superset of $C$ must be a subset of $K \setminus U$, therefore a subset of $K$, therefore by assumption equal to $C$.
Wihout too much trouble we can now prove:
Let $X$ be a compact Hausdorff space, $Y$ an open subspace and $Z$ a closed subspace. Let $C$ be a connected subset of $Y \cap Z$ such that $C$ is a component of $Y$ and a component of $Z$. Then $C$ is a component of $Y\cup Z$.
Proof:
$C$ is a component of, therefore closed in $Z$, which is closed in $X$, so $C$ is closed in $X$. $Y$ is open in $X$, so $X \setminus Y$ is closed in $X$.
$X$ is normal, so $C$ and $X \setminus Y$ have disjoint neighbourhoods $U$ and $V$. If we take $K = \operatorname{Cl} U$ then $ C \subset U \subset K \subset X \setminus V \subset Y \subset Y \cup Z $ and $K$ is compact.
Starting from the fact that $C$ is a component of $Y$, we now apply the lemma one way to find that $C$ is a component of $K$, then the other way to find it is a component of $Y \cup Z$.