As others have said, conditions can range from summer heat to winter blasts in June. Our trip to Whitney in early July, 2001, was comfortable...but threatening with afternoon t-storms. It never got down to freezing at Trail Camp, but was cool and windy enough from there to the summit that we needed polypro, one fleece layer on top, and our Goretex parkas for the rain/sleet/hail on the way back to Trail Camp.
Of course, then there was our 1996 <a href="http://www.mtritter.org">Mt. Ritter</a> trip, when we walked into the teeth of a blizzard at 11,000' before turning back. That year, discretion was definitely the better part of valor and backing down from serious winter conditions was the right thing to do.
Basically, watch the forecasts the days before you head up Whitney and be prepared to make sensible decisions. Watch for developing conditions (look at my summit panorama in the 2001 trip report) and be ready to head for lower ground
before the lightning starts!
Also, be aware that, when you're on the Whitney summit, you still have a couple of miles of exposed trail at or near the ridgeline before you get back to the switchbacks, so it can take a significant amount of time to get to relative safety from lightning if you let yourself get caught up there at the wrong time.
Basically, in early June, I'd plan on slightly sub-freezing temps overnight at Trail Camp. I carry a pair of fleece mittens, stocking cap and fleece pullover out of habit. Long johns and lightweight nylon pants, plus a pair of rain pants and a Goretex parka should be quite adequate for anything except the really unusual (like our '96 Mt. Ritter trip). In those cases, remember the prime rule of mountaineering, and its first corollary:
1. The summit is optional. Returning in one piece is mandatory.
1.a) The mountain will still be there next year.
