Recent work has identified a dynamical squeezing phase transition in power-law interacting bilayer XXZ spin models, separating a fully collective phase with Heisenberg-limited squeezing from a partially collective phase with universal critical scaling. Here we test and establish the universality of this transition along two qualitatively different microscopic axes: lattice geometry, by studying square, triangular, and honeycomb $2D$ bilayers as well as $1D$ ladders, and a symmetry-preserving rescaling $\lambda$ of the interlayer couplings relative to the intralayer ones. Combining a Bogoliubov instability analysis with discrete truncated Wigner simulations, we find that the transition persists across all four lattice geometries and over a wide range of $\lambda$ with critical exponents consistent within error, providing strong evidence for a genuine non-equilibrium universality class. The Bogoliubov theory recovers the previously identified scaling $a_Z^{*}\propto L$ in the long-range interacting regime $\alpha <d+2$, and yields an analytical scaling $a_Z^{*}\propto L^{2/(\alpha -d)}$ for the critical aspect ratio with system size for $\alpha >d+2$, with $\alpha$ the power-law exponent in dimension $d$. This uncovers a previously unrecognized sub-linear regime for short-range interactions. By tuning $\lambda$ we vary the interlayer coupling strength at fixed layer spacing, demonstrating that the dynamical transition can be driven purely through interaction engineering without modifying the underlying geometry. These findings provide a versatile route toward controlling entanglement generation in Rydberg-array, polar molecule, and trapped-ion platforms with applications in quantum sensing and simulation.