I don't think choice is needed to do the following...

Suppose $X=\{x_i:i<\omega\}$ is a countable  *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 $C_n$. Contradiction.