I have heard that wrapping tin foil (aluminium foil) around the base of the plant will help stop them from getting in the plant, and sorry but I can't answer your other questions (I'm also new to gardening), hope this helped!

How did you cut the stem: lengthwise or did you cut it crosswise? I would imagine you slit lengthwise in order to get the borer and yet still let the plant have a possibility of growing. If you slit the stem with the grain (lengthwise) then the top of the squash plant may still be green and alive. If so, then I'd recommend that you pile soil over the damaged area and, yes, water it, in hopes of getting the vine to heal and grow some more roots at the damaged place, If you cut the vine off (crosswise) then, no, it is unlikely that anything will sprout from the roots.

In what region do you live? You may have time to re-plant, which would good practice, even if you hadn't had the borer problem (presuming you have the space, etc.)

Mrbenjabib may be new, but his advice is good. I have always wondered if the tin foil really helps. But my limited experience with it seems to indicate that it does. I have even found good results with simply laying the tin foil flat on the ground, around the base of the plant.

One other observation: don't feel like you've messed up if your plant does not recover. I've grown squash most of my life and found that once a borer hits a vine it is usually not going to recover, no matter what I do. Still, I always think it's worth the try.

George
Tahlequah, OK