on a normal year there will be no problem going through the sierras in early june. yes there will be snow. no you wont NEED an ice axe or crampons, BUT they might prove usefull if you are not experienced in snow hiking. if you go in early june dont expect to do more than 15 miles a day comfortably unless you are in really good shape especially if you are only doing the jmt. the pct hikers have an advantage because they have alreay been hiking for a month and a half or so and they are already in shape and used to 20-25 mile days.
dont worry about it to much. and dont listen to alot of the hubub that alot of people tell you because most of them are exagerating about the difficulty and danger level. that was the case on the pct this last year, everybody was saying "dont go over fuller ridge, its way to dangerous and to much snow" so most people road walked around. i went over fuller ridge in the early season with high snow and it was fine. there was no real danger to be worried about. it was the same thing with mt. baden powell, alot of people didnt go up and over the mountain because others said it was to dangerous, but again these claims were highly exagerated. it really wasnt that bad.
have a great hike and just be careful out there, remember this : only you know your own limits, and be sure to remind yourself of your limits so you dont get in trouble out there.
TYVEK
TYVEK