Benefits of starting freshman year of college in the summer?

I applied to duquesne university (my dream school) for the fall 2017 semester. They later said they would only accept me if I start this June. Problem is, is that I have plans. My family and I payed for a vacation (our first one, since we could never afford one before) and I committed to a youth group where I will travel to Haiti to volunteer. I would feel awful if I told them I couldn’t do it. I also don’t want to miss out on the summer before college, all of my older friends tell me it was the best few months of their life. Do college expect you to not have any plans during the summer?? Am I being too picky? What are the benefits of starting in the summer?