The San Juans may still have some snow on the higher elevations in June-- so I would try to find more information about the ground conditions closer to the date and be prepared for that possibilty.
As for clothing, I would take your Cap 1 and R1, along with the down sweater and a rain jacket for your upper body. If you have one, then add a windshirt also. Lower half, just some Cap 1 or Cap 2 weight baselayer pants should be all you need for June/July. If you have rain or wind pants then you may want to throw those in your pack also, but if not then don't worry about it. Don't forget a warm hat and warm gloves, along with a sun hat too.
I don't use a Steripen of any kind, but due to what I have heard about them, you may also want to take some MicroPur tablets as back-up.
I would leave the Camelback and Kleen Canteen at home and just take a 2L collapsable, wide-mouth Nalgene bottle with one wide-mouth Gatorade bottle to use with the Steripen.
I would switch out the Jetboil in favor of an alcohol stove.
Food-- general rule of thumb is 1.5-2lbs of food per person, per day, but it really depends on your cooking tyle and eating habits. Some people like more gourmet meals, others just freezer bag meals, energy bars, nuts, dried fruit, GORP etc.
Pads, I get by with a Ridgerest or Z-lite. I personally believe the NeoAir is overpriced for its usefulness. I purchsed one, ended up getting cold in the low-to-mid 30s and now it is back at REI, but as always YMMV.