Nope - any wire will do. Shielded may be better, but many companies now don't even bother with that. Any electronic 'hookup wire' will do. 22-20 guage is ideal, 24 will work, but starts lacking in structural integrity, 18 or thicker and it gets bulky and hard to solder to pot casings properly.
I recently used some scrap offcuts of console hookup wire (mic/line, but the super thin stuff for patchbay etc wiring looms, ignored one conductor) to replace switch to output jack wiring on a bud's Squier Strat (which had the crappiest wiring in it - just crumbled when I touched it). For non-shielded, thin guage speaker wire is just as good as anything. When I did my kitchen the new lighting fixtures came with 1 meter extra length of wire per conductor and I use that for stuff too because it's surprisingly nice.
Bridge ground can go to any ground point, yes. However, it is best to use a star ground scheme and have all ground connections come to a central point (usually one of the pot casings is the most convenient), and only 1 wire to the jack from there. That said, I have never had a ground loop problem in a guitar, even when I tried that one time just to see.