Ysmtrnr, the Mountaineers Route is the NON-trail route up the North Fork Lone Pine Creek. It requires the same day use permit as the main trail (On could hike either route), but requires its own permit to hike there and spend the night.
It is shown on this map
as if there is a trail, and part of the line is completely wrong.
ZoeClimbs: I think by mid-June, the weather will be settling back to summer conditions, and snow will be melting out. You will likely cross some snow on the west side, but east side should be drying out by then. I wouldn't carry the microspikes if I were going, but then I am pretty sure-footed.
Doug at the Whitney Portal Store reported here
that there is snow above 11k elevation, a few people are making it to the summit, but most are turning around at Trail Camp or just above. Yesterday saw 8" of snow high on the mountain. The wintery weather is convincing most to turn back.