False, no such thing happens.  This is a myth.  Yes, prairie dogs are vermin.  My family pastured our cattle in the summer on the north rim of the North Platte valley on land that had a couple of prairie dog towns.  No cattle ever broke their leg, let along died from falling into a prairie dog hole.  My father's only warning was to not gallop my horse through either town to avoid the possibility of my horse stumbling.  Horses and cattle are different critters.