  Spin squeezing -- a central resource for quantum metrology -- results from
the non-linear, entangling evolution of an initially factorized spin state.
Here we show that universal squeezing dynamics is generated by a very large
class of $S=1/2$ spin Hamiltonians with axial symmetry, in relationship with
the existence of a peculiar structure of the low-lying Hamiltonian eigenstates
-- the so-called Anderson's tower of states. Such states are fundamentally
related to the appearance of spontaneous symmetry breaking in quantum systems,
and they are parametrically close to the eigenstates of a planar rotor (Dicke
states), in that they feature an anomalously large value of the total angular
momentum. We show that, starting from a coherent spin state, a generic
$U(1)$-symmetric Hamiltonian featuring the Anderson's tower of states generates
the same squeezing evolution at short times as the one governed by the
paradigmatic one-axis-twisting (or planar rotor) model of squeezing dynamics.
The full squeezing evolution is seemingly reproduced for interactions decaying
with distance $r$ as $r^{-\alpha}$ when $\alpha < 5d/3$ in $d$ dimensions. Our
results connect quantum simulation with quantum metrology by unveiling the
squeezing power of a large variety of Hamiltonian dynamics that are currently
implemented by different quantum simulation platforms.
