  Controlling the quantum many-body state of arrays of qudits, possessing a
large local Hilbert space, opens the path to a broad range of possibilities for
many-particle entanglement, interesting both for fundamental quantum science,
as well as for potential metrological applications. In this work we
theoretically show that the spin-spin interactions realized in two-dimensional
Mott insulators of large-spin magnetic atoms (such as Cr, Er or Dy) lead to
scalable spin squeezing along the non-equilibrium unitary evolution initialized
in a coherent spin state. An experimentally relevant perturbation to the
collective squeezing dynamics is offered by a quadratic Zeeman shift, which
leads instead to squeezing of individual spins. Making use of a truncated
cumulant expansion for the quantum fluctuations of the spin array, we show
that, for sufficiently small quadratic shifts, the spin squeezing dynamics is
akin to that produced by the paradigmatic one-axis-twisting (OAT) model -- as
expected from an effective separation between collective spin and spin-wave
variables. Spin squeezing with OAT-like scaling is shown to be protected by the
robustness of long-range ferromagnetic order to quadratic shifts in the
equilibrium phase diagram of the system, that we reconstruct via quantum Monte
Carlo and mean-field theory.
