Jeanne,
Welcome to QJ!
Well, you have certainly picked what CAN be a #1 bug week in the park. However, having said that, actual conditions you experience will depend upon lots of factors. I've been in the park many times that very week & have experienced widely varying conditions. One time we almost went without using bug dope of any sort. Heck, Big Ed, a rather large friend of mine - prime bug meat in any season - actually slept outside of any tent, under our tarp, EVERY night of an 8 day trip during a late June trip!
In other years, even Big Ed might have been exsanguinated....
Of prime importance is the question: how wet have conditions been during the previous three weeks? IF wet, plan on heavy bugs. Certainly bring bug dope & probably head nets, too, though you would mainly need nets during portages, in the woods, or in passage through closed-in narrow waterways & swampy areas. Even when it has been wet, generally the skeeters are not much of a factor (except in those places mentioned) EXCEPT at the point their "dinner bell" rings. Last year, during that very week of the season, their dinner bell was ringing at approximately 9:34pm (plus or minus 3-4 minutes). You could hear the swarms in the distance, revving up their engines, twenty minutes in advance of their onslaught. Then, when the bewitching hour/minute struck, we quickly donned the headnets OR headed for the tents 'cause they were ON us & THICK!
While bug-swarming conditions may mitigate during the dark hours, you can count on them being pretty pesky for the first hour or so around dawn, too.
My remarks pertain to skeeters, not black flies, ticks, no-see-ums, or canoe flies... each of which respond to their own sets of dynamics.
Ticks, for instance, SEEM to have slackened off, somewhat, by the end of June.
As for black flies, they tend to be worse earlier in the season, too, & particularly near the moving waters where they hatch.
Canoe flies (a.k.a. "ankle biters" with the six inch teeth) can get bad by late June, but my experience has been that they get worse a few weeks later on. Roll your socks over your trousers, rubber band them down & you've done most of what you CAN do with these beggars... though I'd be tempted to use a Machine Gun, if one was handy.
No-see-ums are rarely a problem BUT, when they are, the best remedy I have come up with is to "move camp."
Except for ticks, you can alleviate a lot of your bug problems around camp by selecting a spot likely to catch a breeze. Of course, THAT has its downsides, too, when fierce summer storms blow in.
My advice? Bring the bug nets for portages, paddling where the breezes don't blow, & for the hours around dusk & dawn. Find airy campsites. Carry bug dope with DEET. Also, while the park desperately needs moisture right now, you might want to stop the rain dance about a month before you head into Quetico. The conditions you experience can be extreme depending upon your route & the recent rains.
Jimbo