Ordering and segregation properties of Fe-Ni alloys and nanoalloys are investigated by means of Metropolis Monte Carlo (MMC) and molecular dynamics (MD) simulations. The model is based on an embedded atom potential which, according to thermodynamic integration, only stabilizes those phases that are observed experimentally. This stability is confirmed by MMC and the same phases are found stable in truncated octahedral nanoparticles containing no more than 201 atoms. At given composition, Ni segregates at {100} and nanoparticle surfaces on the Fe-rich side of the phase diagram, Fe segregates at intermediate compositions and no significant trend is predicted on the Ni-rich side. A BCC to L1(0) transition is observed to occur at a Ni fraction close to 0.32, both in bulk alloys and in nanoparticles. The transition gives rise to a change in the nanoparticle aspect ratio by a factor 2(1/2). Using MD, by varying temperature, it was possible to monitor a BCC to FCC transition in solid solution nanoparticles reversibly.