If it was me, I wouldn't be fussed about the weekends. For 4 park days, I would pick one of the following:
June 26-29
June 26-27 and June 29-30
June 27-30
June 27, June 29-July 1
June 28-July 1
June 29-July 2
Reasons: Grad nights are over. So-Cal blockouts have kicked in. Park hours should be at their maximum. Weather could be very nice.
Why skip June 28? In my experience, during the last few years Sundays generally feel more crowded than Saturdays. If you have the opportunity to have a skip day but would like to brave a weekend day, pick Saturday over Sunday. Weekend days are not to be entirely feared/avoided though. But maybe that's just me. Why pick June 28-July 1? You need to do four straight days and are more concerned about the beginning of July 4 crowds than Sunday crowds. Reason to possibly hang in there until July 2 - if Disney does what they have done for the last few years, that could be the first night with the July 4 fireworks before crowds hit their full holiday spikes. Then again, with the new fireworks show this year and all the excitement that is expected to generate, maybe they won't bother running the July 4 fireworks on multiple nights. Anyway, just my two cents of what to consider.