We establish interior regularity and optimal growth estimates for sign-changing minimizers of the \( p \)-singular or \( p \)-degenerate quasilinear Alt-Phillips functional throughout the full range of \( 1<p< \)\( \infty \) and of the nonlinearity power \( 0<\gamma< p \)\( \). In addition, we obtain local finite perimeter and density estimates, from which we deduce the local \( (N-1) \)-rectifiability of the reduced and two-phase free boundaries and the local finiteness of their \( (N-1) \)-dimensional Hausdorff measure for a restricted range of \( \gamma \).