The Winds, Sierras, Uinta's, etc may have an abundance of bugs
Not sure if the Sierra are quite as bad, but that's definately true of the Winds and Uintas, unless they are still buried completely in snow (unlikely so far this year since it's been a dry year).
Options I thought of would be Cascades: South Sister/St Helens/Mt Adams, Olympics,
There's usually a lot of snow that time of year, but it's a good time to climb if you know how to use an ice axe.
The lower end might be hot, but the upper end and Buckskin Gulch would be nice with the water and shaded canyons.
circumnavigation of Navajo Mt on the North and South Rainbow trails (Rainbow Bridge)
It would be hot in places, but there are plenty of swim holes and waterfalls around so it would be a good time of year to go. Also, consider combining Aztec Creek/Frobidding Canyon as part of the loop rather than coming up the South Rainbow Trail. Aztec Creek would be perfect at that time of year and is very spectacular.
Dark Canyon is perfect in June, at least below Black Steer Canyon. With all the swim holes and waterfalls, June is perfect. The best section of Dark is between Lost and Youngs Canyon.
The Maze is nice, but skip it in June. It's way too hot then and there isn't much water around.
I would skip the Grand Canyon in June as well. If you are in that area in June, head to the canyons around Sedona such as West Clear Creek, Wet Beaver Creek and West Fork Oak Creek.
June is perfect for many of the routes. A few are Boulder Creek, Death Hollow and 40 Mile Canyon. All have swimming and June is the best time to go.