 Solution

Interpret the source's $\mathbf{J}$ as the dimensionless angular-momentum operator, with $J_z|J,m_J⟩=m_J|J,m_J⟩$. If angular momentum is written with physical units, the magnetic-moment formula instead contains ${\mathbf{J}}_{\mathrm{phys}}/\hslash$; this is a convention, not an extra factor in the requested energy.

The field fixes the quantization axis, soThere are $2J+1$ Zeeman levels equally spaced in energy by $g{\mu }_{B}B$. For positive $gB$, the state with largest positive moment has lowest energy. The given sign defines $\mathbf{J}$ through the specified moment relation; inserting an additional electron-charge minus sign into it would change the source convention.