I recently planted a set of bare root Virginia Bluebells, and some squirrels dug them out. If I were to plant another set, would laying hardware cloth over the roots prohibit growth?
Virginia Bluebells are planted with the crown at the soiil surface, not deep like a bulb. With common names, sometimes multiple plants have the same name; typically, Virginia Bluebells is the common name for Mertensia virginica: https://davesgarden.com/guides/pf/showimage/189886/. Perhaps your Virginia Bluebell is a different plant - a bulb?
If you are planting Mertensia... I assume that by "hardware cloth" you're referring to 1/4" mesh screening, which will be too small of a grid for Mertensia. I would just lay chicken wire on the top of the soil (using ground staples to hold it in place) to keep the squirrels at bay, and then remove the wire and staples when the plants go dormant in June.