The wave functions, energy levels, and oscillator strengths of B²⁺ ions and B⁺ ions are calculated by using a relativistic potential model, which is named the relativistic configuration interaction plus core polarization (RCICP) method.The presently calculated energy levels are in very good agreement with experimental energy levels tabulated in NIST Atomic Spectra Database, with difference no more than 0.05%.The presently calculated oscillator strengths agree very well with NIST and some available theoretical results. The difference is no more than 0.6%. By using these energy levels and oscillator strengths, the electric-dipole static polarizability of the 2s_1/2, 2p_1/2, 2p_3/2, and 3s_1/2 state and static hyperpolarizability of the ground state 2s_1/2 for B²⁺ ion, as well as electric-dipole static polarizability of the 2s^{2 1}S₀ state and 2s2p ³P₀ state for B⁺ ion are determined, respectively. The polarizability of the 2p_1/2 state and 2p_3/2 state of B²⁺ ion are negative. The main reason is that the absorption energy of the 2p_1/2,3/2 → 2s_1/2 resonance transition is negative. The contribution to the polarizability of the 2p_1/2 state and 2p_3/2 state are both negative. For the tensor polarizability of the 2p_3/2 state, the main contribution from the 2p_3/2 → 2s_1/2 transition and 2p_3/2 → 3d_5/2 transition are 2.4963 a.u. and –0.2537 a.u., respectively, and the present RCICP result is 2.1683 a.u. The largest contribution to the hyperpolarizability of the ground state 2s_1/2 originates from the term of \alpha ^1\beta _0 . The electric-dipole static polarizability of the 2s^{2 1}S₀ state and 2s2p ³P₀ state of B⁺ ion are 9.6220 a.u. and 7.7594 a.u., respectively. The presently calculated blackbody radiation (BBR) shift of the 2s2p ³P₀ → 2s^{2 1}S₀ clock transition is 0.01605 Hz. This BBR shift is one or two orders of magnitude smaller than that for alkaline-earth-metal atom.