it4: Check the
Mt Whitney Weather page for weather links in the week leading up to your hike. (The weather page link is always in the Feature Topics box just below the webcam pictures on the left.)
On that page, we have links to DMatt's weather, and the NOAA 7-day forecast for the summit area. So in a couple of clicks, you can see the weather at any time.
It is quite likely you will have excellent weather in June.