The ground and a few excited states of the boron atom in external uniform magnetic fields are calculated by means of a 2D mesh Hartree-Fock method for field strengths ranging from zero up to 2.35×10⁹ T. With increasing field strength the ground state of the B atom undergoes five crossovers involving six different electronic configurations which belong to three groups with different spin projections S_z = -1/2,-3/2,-5/2. For weak fields the ground state configuration arises from the field-free 1s²2s²2p_-1, S_z = -1/2 configuration. With increasing field strength the ground state involves the four S_z = -3/2 configurations 1s²2s2p₀2p_-1, 1s²2s2p_-13d_-2, 1s²2p₀2p_-13d_-2 and 1s²2p_-13d_-24f_-3, followed by the fully spin-polarized S_z = -5/2 configuration 1s2p_-13d_-24f_-35g_-4. The latter configuration forms the ground state of the boron atom in the high-field regime γ>8.0251. Analogous calculations for the B⁺ give a sequence of the four following ground state configurations: 1s²2s² (S_z = 0), 1s²2s2p_-1 (S_z = -1), 1s²2p_-13d_-2 (S_z = -1) and 1s2p_-13d_-24f_-3 (S_z = -2). The above series of ground state configurations are extracted from the results of numerical calculations for a number of electronic configurations selected according to general energetical arguments.