I really like June in the Sierra. Mosquitoes usually do not get intense until about 4th of July. But, you have to be willing to deal with some walking on snow. If you are a good navigator, the trail disappearing under snow is not a problem. Each year is different. The other difficulty with June are river crossings. You have to be careful to plan crossings of major streams where there are bridges. The southeastern Sierra melts off earlier than northern Sierra, so in spite of the higher elevations, you are less likely to have deep snow to slog through.

The PCT hikers usually are going through the Sierra in June. You should read in detail, some of their journals. 2011 was a high snow year with really tough conditions. 2013 was low snow and few problems.

June in the PNW can be quite rainy. Or it can be just fine. They are having more snow up there this year than we are in the Sierra.

For the Sierra, check this website for snow conditions.

http://cdec.water.ca.gov/snow/current/snow/index.html

IF you decide on the Sierra, check out this website.

http://www.highsierratopix.com/community/