Answer: The problem is most likely fungal or bacterial diseases attacking fruit. Fungicide or bactericide sprays can help, but cultural techniques like not overwatering, using a drip watering system rather than constant overhead sprinkling, and growing vines on a trellis to improve air circulation all will help significantly. I have grown vining squash on a trellis for years and have not had a significant problem with fruit rot. Mulching can certainly help reduce the problem somewhat but is not a 100% solution to this problem.
If you want a spray option, contact your Extension Office and they can recommend an appropriate product.
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