Can a free particle ever emit a particle?

True and true - there is no contradiction here.

The complete answer is contained in the answer to linked question:

This is because energy and momentum are not both conserved if a free charged particle (say, an electron) emits a photon

and the comment to that answer:

Note that if the particle has internal structure, this argument can fail. For example, atomic nuclei emit gamma rays. This is because they have more than one internal state with different energies.

Applying these two principles we arrive at:

  1. A single free electron cannot emit a photon
  2. A single free hydrogen atom with electron in an excited state can emit a photon by spontaneous emission
  3. A single free nucleus can emit a photon again by relaxation of its internal structure.

Note: In this answer I have only considered the dynamical restrictions involved in emitting a single photon. As pointed out in the comments there are many more complex decay processes of either elementary (e.g. tau) or composite (e.g. neutron) particles which may result in the emission of a photon as one of many decay products.