The Wikipedia article on the Perron-Frobenius theorem claims that (without referencing a proper proof):
The key feature of theory in the non-negative case is to find some special subclass of non-negative matrices— irreducible matrices— for which a non-trivial generalization is possible. Namely, although the eigenvalues attaining the maximal absolute value may not be unique, the structure of maximal eigenvalues is under control: they have the form $e^{i2πl/h}r$, where $h$ is an integer called the period of matrix, $r$ is a real strictly positive eigenvalue, and $l = 0, 1, ..., h − 1$.
I have trouble constructing a proof for this and my literature research has unfortunately not been helpful so far.
UPDATE: Does this also hold for all eigenvalues? (and not only for the maximum modulus ones)
Most treatises that cover Perron-Frobenius theorem for nonnegative matrices should contain the relevant proofs. In particular, proofs can be found in vol. 2 of Gantmacher's "The Theory of Matrices" or Horn and Johnson's "Matrix Analysis".