There is a storm hitting the Sierra today, NOAA predicts a few inches of snow should fall on Whitney.

People will likely be going up the slope west of the switchbacks well into May. Last year people were reporting using the switchers by May 23. It is likely this year will be the same.

So yes, take ice axe and crampons.

BTW, it is highly likely you can get a walk-in permit easily any time in June: Unused Whitney Permits - 2013