You can also browse the Current Conditions threads. We keep one each year, and each year's has a link to the prior year's.
You certainly won't need snow shoes. Crampons might be helpful if there is considerable snow. Some years the switchbacks aren't accessible until mid-June. Look at the
Unused Whitney Permits threads -- I've tried to note when the switchers are first being used. When people first use the switchbacks, there is lots of ice and snow, so for a few weeks, crampons, or at least microspikes are helpful.