A: Squash vine borers, a.k.a The Bane of My Existence. If your squash wilt like this, look for orange-ish frass (looks similar to the cheap roe they use on california rolls at lesser sushi restaurants) near the base. Wherever the frass is, the borer is between that point and where the vine meets the dirt. Slit the vine with a knife, pick it (or them) out, and kill it. In the picture below, you can see frass, the slit, and at the top of the slit, a borer.
Here is a smaller borer we picked out and smooshed. They are pretty disgusting.
I had borers in almost all my vines. They say you can mound moist dirt over the slit and the vine may heal (though I'd just pull up a plant that looks as bad as the one above). They also say you can pile moist dirt over healthy stems and they may root at that point. I did both to be on the safe side. I also started more seeds so I can replace the plants if necessary. Supposedly in the deep south, there can be two borer seasons a year, so I may not be able to outsmart them by planting a second set of squash - we'll see!
The thing about suckers is that they aren't just another branch; they're more like another entire tomato plant growing out of the side of your main plant! They are easy to spot at this stage, tougher later - though you can remove them anytime. Tomato plants are actually very forgiving about pruning. I control the height of mine as well as prohibiting suckers.
You can root your suckers in water if you want more plants - I usually let them get about 6 inches long before I cut them off the plant for that purpose.
One caveat about suckers: on some tomato plants, such as yellow pear, the stem on which blooms and babies will form comes from this same area or very close. It's easy to tell the difference, but I don't want you to prune out all your bloom stems by accident!