Description: Presence of flat bands and edge states at the Fermi level in graphene nanoribbons with zigzag edges is one of the most interesting and attracting properties of nanocarbon materials but it is believed that they are quite fragile states and disappear when B and N atoms are doped at around the edges. In this paper, we theoretically investigate electronic and magnetic properties of boron-carbon-nitride (BCN) nanoribbons with zigzag edges where the outermost C atoms on the edges are alternately replaced with B and N atoms using the first principles calculations. We show that BCN nanoribbons have the flat bands and edge states at the Fermi level in both H_2 rich and poor environments. The flat bands are similar to those at graphene nanoribbons with zigzag edges, but the distributions of charge and spin densities are different between them. A tight binding model and the Hubbard model analysis show that the difference in the distribution of charge and spin densities is caused by the different site energies of B and N atoms compared with C atoms.