Swell! Thank you all so much!
If I am understanding correctly, it seems like most western trails will still be snowy in early June. However, I am going to research a bit more about the lower elevation trails.

How about the AT? or other national trails on the east coast?
My buddies and I want to end up in Dover, DE around June 19th for Firefly Music Festival (one of my friends is working it). And from what I have found out so far, there seems to be less snow and other barriers to starting in June (however, with the current storms all over the country, I am sure we are in for a weird backpacking summer).

Thanks again for all the advice!
-J