  Within a self-consistent framework of q-deformed Heisenberg algebra and its
equivalent framework of q-deformed boson commutation relations, which relate to
the under-cutting phenomenon of Heisenberg's minimal uncertainty relation,
special q-deformed squeezed states are constructed. Besides the similar local
maximum squeezing as the one in the undeformed case, new strong squeezing
appears when the amplitude of the related coherent state increases to large
values. A critical phenomenon appears at a large value of the amplitude: the
variance of one component of the quadrature of the light field approaches zero,
but the variance of the corresponding conjugate quantity remains finite, which
is a surprising deviation from Heisenberg's uncertainty relation. The
qualitative character exposed by this q-squeezed state may provide some
evidence about q-deformed effects in current experiments.
