  We present a hybrid optomechanical scheme to achieve dynamical squeezing of
position quadrature of a mesoscopic mechanical oscillator, that can be
externally controlled by classical fields. A membrane-in-the-middle set up is
employed, in which an atom in $\Lambda$ configuration is considered to be
trapped on either side of the membrane inside the cavity. We show that a
considerable amount of squeezing (beyond the 3 dB limit) can be achieved that
is not affected by the decay of the cavity and the spontaneous emission of the
atom. Squeezing depends upon the initial preparation of atomic states. Further,
a strong effective coupling between the atom and the oscillator can be attained
by using large control fields that pump the atom and the cavity. Effect of
thermal phononic bath on squeezing is studied in terms of the squeezing
spectrum. The results are supported by the detailed analytical calculations.
