We theoretically explore nonlinearities of ferromagnets in microwave cavities in the classical and quantum regimes and assess the resources for quantum information, i.e., fluctuation squeezing and bipartite entanglement. The (semi)classical analysis of the anharmonic oscillator (Duffing) model for the Kittel mode when including all other magnon modes, reveals chaotic and limit-cycle phases that do not survive in quantum calculations. However, magnons with nonzero wave numbers that are driven by the Suhl instability of the Kittel mode, form a genuine limit cycle. We subsequently compute bounds for the distillable entanglement, as well as entanglement of formation for the bipartite configurations of the mixed magnon modes. The former vanishes when obtained from a covariance matrix, but can be recovered by injection locking. The predicted magnon entanglement is experimentally accessible with yttrium iron garnet samples under realistic conditions.