There is no expression for spin in the standard theory of electromagnetism. The subject is in fact avoided. For example the only reference to electromagnetic spin in Jackson Classical Electrodynamics is in problem 7.27. The main text does not mention it.
The total angular momentum, including spin, is correctly given by your equation, which has the form of angular momentum. So what is happening? Jackson's p7.27 gives a clue. The Noether AM distribution belonging to the standard, gauge invariant Lagrangian contains a spin and an orbital AM density contribution. However, only their sum is conserved. Also the energy-momentum distribution is asymmetric. None of the Noether current distributions are gauge invariant. The solution to keep everything gauge invariant is to replace the Noether energy-momentum (EM) distribution with the Belinfante distribution. This leads to your expression of AM. The spin contribution is no longer explicit.
I published a paper that proposes an alternative. It also discusses a paradox that arises when you try to understand Beth's experiment, mentioned by @rob, with your expression for $J$. My solution is incompatible with the principle of gauge invariance. Instead it accounts for any physical situation with a unique electromagnetic potential.