You have three trends all competing in June.
1. APs and their blockouts, along with the new World of Color opening in May or maybe June
2. Grad nights
3. Schools ending and transition to peak summer hours
I would give greatest weight to #1. Go when the AP are blocked. See "Understanding AP Blockouts and Impact on Crowds"
http://www.disboards.com/showthread.php?t=2365983
Then I would at least pay attention to Grad nights. See "The Scoop on Grad Nights at DLR in May and June" -
www.disboards.com/showthread.php?t=2298400
Finally, as DLR goes from early to late June the out of school crowds will come. But the park hours will be longer too. Usually around mid-June DLR goes to max hours and max daily entertainment.
But even more than all of the above is your friends' willingness to get to the gates early and take afternoon breaks. If they are like most people and come late to the parks and stay through the heat of the day and longest lines, all the best efforts to choose good days will be for naught.
So, if they can get to the gates early and use good touring strategies I would say the best 4 consective days are June 23-26, all AP blockouts and one grad night on June 24.
If they cannot get to the gates early or understand the value of afternoon breaks, I would say June 9-12. June 10-12 are blocked out, June 10 is a grad night, but the parks will likely have reduced hours on June 9-10. See "Historical DLR Park Hours, Entertainment and Refurbs" -
www.disboards.com/showthread.php?t=2291124