The two mountain passes out of Eugene are Willamette Pass (southeast) and McKenzie Pass (northeast). The old McKenzie highway will likely be closed into June or July (maybe open June this year), but the Santiam Pass cutoff from McKenzie is open year round.
Bugs will probably be pretty bad everywhere except maybe the coast (50 miles west of Eugene) in June.
Lots of the Diamond Peak wilderness area (on the south side of Willamette Pass) may well be open to relatively snow-free hiking by this June.
I don't know how much snow there will in up toward the Green Lakes basin north of Willamette Pass in the Three Sister's wilderness, but it would be a beautiful area to go if it is open. Call the ranger stations to get better ideas on when trails might be clear.
Also, there is a fair amount of hiking around the Waldo Lake area (down the hill toward Eugene from Willamette Pass) that might be lower elevation and more horrid bugs.
Women, well, my wife would say to make sure there is good bug protection (a full net enclosed tent, not a tarp), privacy (a tent, not a tarp), good food, low miles, and good weather. In other words, keep and eye on the 10 day weather forecast, and plan accordingly with a willingness to bag the trip for something else if the weather is bad.
That all being said, you can always make the trip a test! I sometimes wish I'd stuck to my guns and insisted on falling in love with a woman that passed all my tests in that she loved cooking glorious meals, would marry me on the top of a mountain, ski, climb, and backpack with me regardless of weather and etc. Alas, our heart seldom follows our ideals, and in the end, it's probably just as well.