I don't think choice is needed to do the following...
Suppose $X=\{x_i:i<\omega\}$ is a countable non-degenerate continuum (connected compact Hausdorff).
Let $U$ be a open set with $x_0\in U$ and $\overline U\neq X$.
Let $C_0$ be a connected component of $X\setminus U$ intersecting $X\setminus \overline U$.
Then $C_0$ is a non-denenerate continuum.
Assuming non-degenerate continua $C_0\supseteq C_1\supseteq ... C_{n-1}$ have been defined, let $x^*$ be the element of $C_{n-1}$ with least subscript.
Let $U$ be an open set with $x^*\in U$ and $C_{n-1}\setminus \overline U\neq\varnothing$.
Let $C_n$ be a connected component of $C_{n-1}\setminus U$ intersecting $C_{n-1}\setminus \overline U$.
Continuing in this manner, we construct a nested sequence of non-empty compact sets. Their intersection must be non-empty. But on the other hand we ensured each point of $X$ is eventually not in $C_n$.