Who is Captain America to Wolverine? Absolutely nobody. Cap is an American, Wolverine is a Canadian. Wolverine was around and doing his own thing before Steve was ever born. Wolverine has more experience with the Phoenix than any Avenger, so why would he trust their judgement over his own? Wolverine makes a career out of telling the group leader to go screw.
Wolverine only joined on the raid because it's his job to "protect the school at all costs." I'm pretty sure the first he thought about when Cap arrived on campus was JGsHL was about to become concentration camp governed by O*N*E, S.HI.E.L.D, the Avengers or some other organization.