Every particle has (or can have) an antiparticle. Sometimes, it is even his own antiparticle.
An antiparticle $D'$ is (easily said) defined as a particle $D$ after a CPT-transformation. CPT-Symmetry is believed to be a fundamental concept of physical nature.
A CPT-transformation is a complete changing of observables of a particle.
The C stands for charge changing, the P for parity and the T for time, so you just invert these values and get a new state for the particle which is still described by the same field equations.
A neutron has no charge, but it has other variables which are (of course) invariant under CPT-transformations. Also, the neutron is not an elementary particle, it is made of two up-quarks and one down-quark, an anti-neutron is made of two anti-up-quarks and one anti-down-quark. These quarks have charges which can be CPT-transformed.