Answer: Squash vine borers tunnel in the vines, and can be a devastating pest problem. Some gardeners have decent success with splitting the vine lengthwise at the spot of attack and removing the borer. The vine usually heals quite well. Others have reported success with injecting B.t. (an organic caterpillar killer) into the vines when borer damage is first seen. I suspect the injection of insect controlling nematodes may also be a possibility.
I have not found the vine borer to be a problem in late summer and fall in the south. Try replanting in August for a fall harvest.
A final thought...If you go out early in the morning, starting just before your plants begin to bloom and set fruit, you will see the "momma" borer sitting on the leaves of plants. She is a black and orange moth about 1" long, but looks more like a wasp in appearance as her wings lie parallel to her body. They are a bit sluggish in the cool of the morning. If you move in slow you can nab the culprit before she lays so many eggs!
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