There's a tendency for anyone who learns information theory before statistics to think that information theory is tremendously relevant - but no, rephrasing maximum likelihood or Bayesian methods in information theory terms may sometimes be slightly helpful in thinking about them, but doesn't really add anything fundamental.
And no, there's nothing particularly special or interesting about distributions that maximize entropy subject to some (generally arbitrarily selected) constraint.
Wow, some two heavyweight opinions there. Care to elaborate?