No.
First of all, quantum mechanics requires that electron orbits are confined to specific quantized
distances from the nucleus. No matter how much pressure is applied, quantum mechanics
still rules.
The electrons in an atom start out filling the electron shells beginning at the lowest, most stable
energy configuration allowed. The Pauli exclusion principle forbids pushing more electrons into
a lower shell if that shell is already filled, no matter how much energy is pumped into the atom.
In fact, pumping more energy into the atom would tend to raise the orbit of an electron to a higher
more distant shell, rather than decrease it. But even then, the energy must be in a form that the
electron can absorb. It has to be in the form of a quantum of energy of just the right frequency for
the electron to absorb it, or the electron will not budge. So, applying pressure to an atom would
simply increase the kinetic energy of the electrons and raise them to higher orbits, if anything at
all.
Now there is such a thing as collapsed matter. This is an extreme case where gravity provides the
necessary extreme pressure in the form of a neutron star. But even here, the electrons are not
squeezed into lower orbits. The exclusion principle still prohibits the electrons from taking lower
orbits, so the atoms simply lose their electrons which become free, and the atom takes on the form
of a naked nucleus.