Make all your lines. Make them Movie Clips, and give them all unique instance names. Put them all on the stage where they should appear.
Looks pretty messy right? No worries... a little AS and you will be golden.
First, in the first frame put this:
Just do that for each line. This will make them invisible when the movie loads. Obviously change "instanceName" with the names you gave the lines.
Next on each county AND button, in the actions panel put this:
This will make the line visible when they hover over the country/button, and invisible when they are not hovering over.
Hope it helps