 Solution

Let $D$ be this union of rational-coordinate horizontal and vertical lines. A point $(x,y)\in D$ has either rational $x$, in which case it moves vertically to $(x,0)$ then horizontally to $(0,0)$, or rational $y$, in which case it moves horizontally to $(0,y)$ then vertically to $(0,0)$. Every segment remains on a line with a rational fixed coordinate.

Thus $D$ is path-connected and connected. It is not necessary for both coordinates to be rational.