It could be a number of things. It could be natural withering of old leaves. Or it could be a nutrient deficiency, or even a disease or insect problem.

If you are fertilizing the squash, it probably isn't a nutrient deficiency. If you're not fertilizing it, then you might want to add some.

It could be vine borers. Check the stems near the root to see if you can find insect damage. If so, then you can try to pick out the borers and then spray with sevin.

It could be a disease also, and you may want to spray the plants with an antifungal (captan) along with an antibacterial (streptomycin). Streptomycin is often sold as fireblight spray.

If it is disease, be aware that many diseases are spread by insects, so you should spray for bugs as well. But when spraying insecticides on squash, you need to spray early in the morning before the blossoms open or in the evening after they close, so that the bees don't get harmed.

This is my first year in planting Atlantic Giant Pumpkins. I live in the South Central part of Virginia and planted the AG seeds directly into the soil June 16th, which I fertilized with 7 bags of cow manure with leaf mulch underneath. It is now August 30th and one of the pumpkins is as big as a watermelon. I also got small umbrellas shading the largest pumpkins, just kind of wondering how much longer they got to grow and how much larger will they get. I use miracle grow on the pumpkins twice a week too.

Also I am already planning for next years crop of Atlantic Giants, I can see why people get hooked on growing giant pumpkins.