  [Abridged] Recently, the production of primordial black holes (PBHs) and
secondary gravitational waves (GWs) due to enhanced scalar power on small
scales have garnered considerable attention in the literature. Often, the
mechanism considered to arrive at such enhanced power involves a modification
of the standard slow roll inflationary dynamics, achieved with the aid of
fine-tuned potentials. In this work, we investigate another well known method
to generate features in the power spectrum wherein the initial state of the
perturbations is assumed to be squeezed states. This approach is technically
straightforward to implement since the Bogoliubov coefficients characterizing
the squeezed initial states, can be immediately determined from the desired
form of the power spectrum. It is known that, for squeezed initial states, the
scalar bispectrum is strongly scale dependent and the consistency condition
governing the scalar bispectrum in the squeezed limit is violated. In fact, the
non-Gaussianity parameter $f_{_{\rm NL}}$ characterizing the scalar bispectrum
proves to be inversely proportional to the squeezed mode and this dependence
enhances its amplitude at large wave numbers making it highly sensitive to even
a small deviation from the standard Bunch-Davies vacuum. These aspects can
possibly aid in leading to enhanced formation of PBHs and generation of
secondary GWs. However, we find that: (i) the desired form of the squeezed
initial states may be challenging to achieve from a dynamical mechanism, and
(ii) the backreaction due to the excited states severely limits the extent of
deviation from the Bunch-Davies vacuum at large wave numbers. We argue that,
unless the issue of backreaction is circumvented, squeezed initial states
cannot lead to a substantial increase in power on small scales that is required
for enhanced formation of PBHs and generation of secondary GWs.
