Every countable subset $A\subset \mathbb{R}^n$, with $n\gt 1$, and indeed, every subset of size less than the continuum, has a path-connected complement. This is because for any two points in the complement, there is a foliation of continuum many paths joining them, and so one most of these paths lies lie entirely in the complement of $A$.
This observation also appears to answer the exercise in your link.
The argument in your final paragraph appears to be conflating conflate $\mathbb{R}^n-B^n$ with $(\mathbb{R}-B)^n$.(\mathbb{R}-B)^n$, but these are not generally the same and they cannot be equal when $B$ is countable.
