Epsilon regularity for minimal surfaces in arbitrary Riemannian manifolds

Deane's comment, which refers to this paper gives the basic answer for minimal hypersurfaces in arbitrary dimensions. (See also Wickramasekera's Annals paper for a recent vast general theory for regularity of minimal "hypersurfaces".)

For higher codimension: the short answer is that $\epsilon$-regularity doesn't work. (At least no one has figured out a sufficient version of it.) This is due to a very real obstruction. A very good summary of the status-quo for the regularity theory of minimal submanifolds is given by De Lellis in this survey article (alternate stable DOI link here); section 5 concerns precisely the question you asked.

The desired bound is correct (and in fact you get a fairly explicit value for $\epsilon$ of anything below $4\pi$) . A proof can be found in these beautiful notes of a course by Brian White (it's Theorem 8.12).