Neutrons are classified as particles, but they are not elementary particles because they are composed of quarks. The discussion highlights that while neutrons and protons consist of three quarks, they still qualify as particles despite their substructure. Neutrinos and dark matter are mentioned as examples of uncharged particles. The conversation also touches on the nature of electromagnetism and its relation to particles. Ultimately, the distinction between elementary and composite particles is clarified, emphasizing that quarks are the fundamental building blocks of neutrons.