  We study the quantum properties of a nanomechanical oscillator via the
squeezing of the oscillator amplitude. The static longitudinal compressive
force $F_0$ close to a critical value at the Euler buckling instability leads
to an anharmonic term in the Hamiltonian and thus the squeezing properties of
the nanomechanical oscillator are to be obtained from the Hamiltonian of the
form $H= a^{\dag}a+\beta (a^{\dag}+a)^4/4$. This Hamiltonian has no exact
solution unlike the other known models of nonlinear interactions of the forms
$a^{\dag 2}a^2$, $(a^{\dag}a)^2$ and
$a^{\dag4}+a^4-(a^{\dag2}a^2+a^2a^{\dag2})$ previously employed in quantum
optics to study squeezing. Here we solve the Schr\"odinger equation numerically
and show that in-phase quadrature gets squeezed for both ground state and
coherent states. The squeezing can be controlled by bringing $F_0$ close to or
far from the critical value $F_c$. We further study the effect of the
transverse driving force on the squeezing in nanomechanical oscillator.
