Your squash plants were infested with a disease called powdery mildew, which can be common in Utah.  There are over 300 different species of powdery mildew, and many of them are host specific, meaning the mildew that you have on your squash, cucumbers, and other cucurbits will not spread to a different type of plant.  The white you are seeing is the mycelium and fruiting bodies of the fungus itself (Sphaerotheca fuliginea). 

Initial infection occurs via airborne spores from neighboring plants.  The disease can develop very quickly once infection occurs, and is first visible on the shaded undergrowth, where leaves are dense, light is low, and humidity is moderate.  The infected leaves will turn yellow, wither, and die.

The best way to manage powdery mildew in the future is prevention.  Once infection has occurred, there is no way to “cure” that plant.  Use powdery mildew-resistant cultivars of squash and other cucurbits.  Examples include ‘sungold’, ‘sunray’, and ‘success’ summer squashes, ‘payroll’ and ‘ambassador’ zucchinis, and ‘table star’, ‘bugle’, and ‘celebration’ winter squashes.  In fall, remove and destroy all infected plant material—do not till the debris from your vegetable garden into the soil.  In the spring, start your plantings in a healthy, rich soil, and use wide spacing.  During the growing season,  improve air circulation around the plants by keeping the area weed-free, and avoid over-head watering and excessive fertilization.

Monitor your plants throughout the summer for infection.  If you notice any spotting or curling of the foliage, or circular white patches on the leaves, remove the leaves or plant.  At this time, start applying preventative fungicide sprays every 2 weeks.  Examples include neem oil and sulfur.  Research has shown that a combination of baking soda and horticultural oil can prevent infection (1 tbs. baking soda and 2.5 tbs. oil in one gallon of water).