June weather in the area is mostly clear and dry. But you need to watch forecasts closely. It is definitely worth planning the climbs, though.
Ice axe and crampons are likely to be quite helpful.
You would need a Whitney day hike permit, but in June, walk-ins are pretty easy to come by, but you would need to stop by the Lone Pine Visitor Center the morning before you hike. ...maybe on your way north to Boundary Pk.
