In myosin II are regulatory and essential light chains calcium binding proteins or sites of phosphorylation?
The answer is both, the regulatory light chains , regulate myosin by phosphorylation which  can be dependent or independent of $\ce{Ca^2+}$.

Medical Physiology:


  The activity of the myosin regulatory light chain is in turn regulated through phosphorylation by $\ce{Ca^2+}$ -dependent and $\ce{Ca^2+}$ -independent kinases.


Source claiming calcium is essential for phosphorylation in tarantula skeletal muscle:


  Contraction is modulated in many striated muscles by $\ce{Ca^2+}$-calmodulin dependent phosphorylation of the myosin regulatory light chain (RLC) by myosin light chain kinase.


Calcium independent phosphorylation of regulatory protein by MLCK4:


  We find that MLCK4 is also expressed abundantly in cardiac muscle, and structural analyses indicate that it is a $\ce{Ca^2+}$/calmodulin (CaM)-independent kinase, in contrast to $\ce{Ca^2+}$/CaM-stimulated cMLCK.