Hey Jib,
When the Mountaineer's route is dry, ie. no snow, it's rated class 3. Which means that you won't need any technical climbing gear of any kind. The route is mostly a very! steep hike. The "climbing" you'll encounter is easy/moderate, although scary at times for some. (When I say "climbing," I'm referring to the times when you'll need to use your arms and hands.) The first climbing will be found at the Ebersbacher ledges about 2/3's the distance to Lower Boyscout Lake. You will have to pull yourself up onto many ledges and over some boulders, all of which is pretty easy. The entire route doesn't get any more difficult, "climbing" wise. The rest of the climbing is up near the summit/notch area, and again, is pretty easy/moderate depending on your fitness level.
When there is snow, obviously the situation changes. At the end of May 2002, I didn't encounter any significant snow all the way to the Notch near 13,000'. This year, at the end of May, I hit snow below Lower Boyscout Lake near 9500'. The snow level varies from year to year so keep a close eye on the trip reports and trail condition reports prior to your departure. The Whitney web cam is an excellent indicator if you know what your looking at...
Also, check out David Gair's post on the message board. He has some really great photos of the route. In June, I'm sure you'll have much less snow than that shown in his photo, but you never know...
Happy hiking(climbing...whatever)