I love June.  Treat your clothing beforehand with permethrin (mainly for ticks), take a headnet and some good spray with DEET, and if the bugs are bad, live with it.

June has a variety of weather, long days, misty cool mornings, and usually doesn't have 90-degree days to sweat in the sun.  The campsites in the BWCA are still fresh, and there are fewer people because many people are wimps about BUGS and stay away until July or August.    ;)

June is the best time to go--although personally I like the second week of June better than the first.