A steady-state mechanical modes squeezing scheme using a ring-cavity optomechanical system is proposed. The system consists of a fixed mirror and two oscillating mirrors. The oscillating mirrors can be regarded as non-harmonic oscillators with cubic nonlinearity. Cavity optical field driven by input laser produces radiation pressure on the oscillating mirrors. The steady-state amplitude of the optical field and mechanical modes in the cavity and squeezing properties of the mechanical modes are investigated. Results show that the steady-state amplitude increases with the increasing of driving power. The mechanical modes can be squeezed in a large range of parameters, and the squeezing can be enhanced by increasing the cubic nonlinearity strength and power of the driving laser, or adjusting appropriately the frequency detuning between the driving field and cavity field. The squeezing is robust to thermal phonon noise.