When they went to rescue Magneto in X-Men: Days of Future Past, why did they take Xavier since he had no real powers and did nothing? He literally couldn't do anything and was a burden on the team since he was just a normal human being at the moment and did not have any special powers. Was there any special reason to take him to the Pentagon?
What's to stop Eric from breaking away from Logan and the others upon being broken out of prison? Quicksilver is just some kid with authority and motivational issues who only helps them because he gets to break into The Pentagon, one of the most secure locations on Earth at the time. No one knew Logan officially until several decades later, plus, why would Eric believe, "Oh, I'm from a horrible future where everything's gone wrong but my mind was sent back in time to my younger body to warn you all of this future so we can work to stop it?" Hank and Eric know one another, but not on a personal level, more like a professional one.
Charles is the only one of the four he shares any kind of close relationship with due to Charles' training to help Eric harness and control his power. They may disagree on the future of Mutant-kind's place within society, but Eric still has respect for Charles even if he disagrees with his methods. Further, they share a common bond through Raven, who has now managed to estrange herself from both Charles and Eric, and manages to be the entire reason the future Logan comes from even happens. Both of them would likely do anything to save Raven, given what they know happened to the Mutants who went with Eric after the incident on the beach in First Class.
Having Charles with them would impress how dire the situation is, improving their chances of his cooporation in their efforts to break him out. They sort of patch things up on the plane ride afterwards, giving them enough reason to work together to prevent the future Logan comes from.
From a more tactical standpoint, Charles knows where Eric is in The Pentagon, and where they'd need to go so they can extract him. Wireless communications were too bulky and loud and obvious in the early 70s, so if any of their plans needed to change on-the-fly they'd need Charles on the ground to help with that.
At the time of the breakout beginning Eric was unaware that Charles had lost his powers. Given that he also no longer had the helmet he had removed from Shaw in First Class he would have to go along with the escape for fear of Charles entering his mind. It is only after Quicksilver saves the day in the kitchen that Charles admits his loss.