Powdery mildew on squash (and almost all other affected plants) is a foliar disease. It doesn't affect the fruit, which looks very clean to me.

It is also not caused by wet weather - water on the foliage actually prevents the spores from germinating. It can occur in late season even in very dry weather as long as humidity is high. Removing any affected foliage and thinning the rest to improve air circulation and reduce immediate humidity is advised.

I took the picture in the morning when there was still dew on everything, hiding the white coating visible otherwise. ... ... We are experiencing high humidity in the air, not much I can do about. Would putting cardboard under the fruit help? Or will it just soak up more moisture whenever it rains ar gets watered?

FWIW, winter squashes can get a waxy bloom that looks white.

Putting something under the fruits might help keep off soil pests that like to nibble, but not sure it would do anything to deter mildew.

I do believe mulching can help prevent diseases like mildew by covering the soil and all the stuff in it, keeping it off plants, among other things.

PM does not affect fruit! Putting down cardboard may help deter soil borne pests but it won't help with the PM. You can reduce the ambient humidity surrounding the plant by doing as I suggested. Also, a deep, slow watering of the root zone can help to equalize humidity in the immediate area and make it less hospitable to the fungus.

Mulching well does help as it deters the soil from drying out too quickly.