he aim of this article is to investigate nonnegativity of the inverse, the Moore--Penrose inverse and other generalized inverses, in the setting of indefinite inner product spaces with respect to the indefinite matrix product. We also propose and investigate generalizations of the corresponding notions of matrix monotonicity, namely, $\circ$-(rectangular) monotonicity, $\circ$-semimonotonicity and $\circ$-weak monotonicity and its interplay with nonnegativity of various generalized inverses in the same setting.