Specifically, it comes from solving the wave equation given the boundary conditions, and those boundary conditions are fixed by the constraint that the square of the wave function (the probability of finding a particle somewhere = 1.
Similarly, atoms can be thought of as a "spherical box", the boundary in this case is a full revolution around the potential well.
The thing is: Not everything is a box, hence the abstract model that uses photons as a shorthand for the behaviour of the EM field in a box isn't fully general, and simply doesn't apply outside of those scenarios.
That doesn't stop the vast majority of people talking about photons (and bosonic particles in general) as if they're individually countable things zipping around in free space.
Neutrons can be counted like this. Photons can't.
Yes, they can. There are detectors that detect individual, free photons, just as there are detectors that detect individual, free neutrons. There are many, many experiments that have been done using them.
If you think you can disentangle the QM properties of the detector from the QM properties of the thing being detected, please write a paper explaining how and collect your well-deserved Nobel prize.
Which, if you are going to take this position, applies just as much to neutrons as to photons. The only evidence we have for either of those being "particles" is such detections. So either you should withdraw your claim that neutrons are particles, or you should admit that photons are. You can't pick and choose.
Composite fermion systems may act as bosons and be "squeezed." Bose-Einstein condensates are an example of this.