Choose such a plane. It contains at most one point from each line in A. So that reduces the problem from three dimensions to two. The two-dimensional problem is this: Given a countable set of points B in a plane, show that the complement of B is connected.
The proof of that is a similar counting argument. Given two points a and b in the plane, there are uncountably many disjoint circular arcs connecting a and b, with the property that any two of them are almost disjoint (in the sense that their intersection consists only of the points a and b). Only countably many of these can contain a point in B. Choose an arc containing no such points. That demonstrates that the complement of B is connected (in fact, path-connected).