Spin of photons and angular momentum of classical radiation field. Are they connected?

They are the same. This is the subject of my favorite classic physics paper: Richard Beth, Mechanical detection and measurement of the angular momentum of Light, Physical Review 50 115 (1936).

Beth constructed a torsion pendulum by suspending a half-wave plate from a fiber, and illuminated the device from below with bright circularly-polarized light. Above this he mounted a fixed quarter-wave plate and a mirror. In the quantum-mechanical picture, each circularly-polarized photon traveling through the half-wave plate exchanges angular momentum $2hbar$ with the pendulum on the way up, has its spin returned to the original orientation during its two trips through the fixed quarter-wave plate, and exchanges another $2hbar$ with the half-wave plate on the way down. By toggling a circular polarizer outside of the vacuum chamber at the pendulum’s resonant frequency, Beth used the angular momentum from this light to cause this macroscopic object to twist.

The angular momentum stored in the classical electromagnetic field predicts the same pendulum motion as the quantum-mechanical explanation; the analysis occupies a substantial fraction of Beth’s paper.

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.