  We propose a method for laser cooling group-II-like atoms without changing
the quantum state of their nuclear spins, thus preserving coherences that are
usually destroyed by optical pumping. As group-II-like atoms have a $^1S_0$
closed-shell ground state, nuclear spin and electronic degrees of freedom are
decoupled, allowing for independent manipulation. The hyperfine interaction
that couples these degrees of freedom in excited states can be suppressed
through the application of external magnetic fields. Our protocol employs
resolved-sideband cooling on the forbidden clock transition, $^1S_0 \to
{}^3P_0$, with quenching via coupling to the rapidly decaying $^1P_1$ state,
deep in the Paschen-Back regime. This makes it possible to laser cool neutral
atomic qubits without destroying the quantum information stored in their
nuclear spins, as shown in two examples, $^{171}$Yb and $^{87}$Sr.
