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I know that strong gravitational fields on the order of neutron stars (at the crust) atoms get compressed so tightly, the empty space between them is significantly reduced and it becomes denser. (http://www.newscientist.com/article/dn16948-star-crust-is-10-billion-times-stronger-than-steel.html#.U4aErvldWSo)

I know that in strong magnetic fields, the atoms polymerize, again becoming denser http://en.wikipedia.org/wiki/Neutron_star

So my question is what happens to atoms in super intense electromagnetic fields on the magnitude comparable to the g-fields and magnetic fields in neutron stars?

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Atoms are bound together by electrostatic forces. The energy is on the order of 10eV (e.g. 13.4eV for for an electron and proton in the ground state of a H atom), and the size is of the order of an Angstrom (Bohr radius is ~0.5 Å). Therefore when electric fields are greater than roughly $$E\sim \frac{10\text{V}}{0.1\text{nm}}=10^{11}\,\text{V/m}$$ then the external field is greater than the binding force of the atom. When this happens you can treat the problem as independent particles in an external electric field (possibly with the electrostatic potential between particles as a perturbation).

If the the field is DC then you will just get constant acceleration of the particles (with the nucleus and electons accelerating in opposite directions). If the field is AC but below the plasmon resonance frequency the charges will oscillate at the field frequency. Above this frequency the field is oscillating too fast to couple to the particles (similar to x-rays passing through matter).

Strong pulses will ionize an atom as the positive nucleus and negative electrons will accelerate in opposite directions. However an AC field (that is linearly polarized) will cause the electrons and nuclei to seperate and then turn around and smash back into each other/recombine. When this happens the energy can be released as much higher energies and is called high harmonic generation which have also created attosecond pulses of light (which hold the record for shortest pulses of light created).

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