The Casimir effect is analogous to gravity in only one way--- it has a negative energy which varies as a power of the distance. In all other ways, it is different. The power-law is different, the cause is different, it is a quantum effect, not a classical effect, and the mediator is the electromagnetic field, not the gravitational field.
Negative energy is not surprising--- it just means that there is an attractive force (and zero potential at infinity), so that the potential energy is less than when the objects are far apart. In this respect, both Casimir and gravity have negative energies. This energy is an energy difference between two configurations.
The total energy in the Casimir system is still positive, it is just less than if you take the two plates to infinity. Similarly, the total energy in a gravitating system is also positive, but when objects are close, it is less than the energy than when they are far. This is the negative energy in gravity.
For the case of Casimir forces, there are two ways to view it. The more primitive way is to consider the fluctuating polarizations in a quantum system that lead to attractive forces. These attractions come in two types, depending on the ratio of the distance to (c times) the orbital period of the electrons (the difference in energy levels converted into a time):
- London forces: when the atoms are significantly closer together than the wavelength of light that they would emit between the first excited state and the ground state, the electromagnetic interaction is instantaneous Coulomb interaction. The Coulomb interaction correlates the charge fluctuations on the two atoms so that they tend to be polarized in the same direction, and this correlated polarization leads to a $1/r^6$ attractive force between points separated by a distance r, or a $1/r^4$ force between two sheets separated by distance r.
- Van-Der-Waals forces: When the atoms are further than the typical wavelength of light, the electrostatic force is no longer instantaneous, and the correlated force is weaker than you get from the statically correlated system. The force falls off as $1/r^7$ for points, and $1/r^5$ for sheets. This calculation is harder than the previous one, but when Casimir calculated the effect, it revealed itself to be more universal than the London-forces, it looked like it doesn't care as much about the energy levels, that it only depends on polarizability of each object separately.
Bohr suggested to Casimir that the universality could be understood because the attraction was due to the correlations in the quantum field surrounding the materials, and that he should calculate the force only by the vacuum energy in the surrounding field modes. This calculation works, and produces the correct form of the Van-Der-Waals force, and explains the universality.
But the effect is not hard to understand--- the Casimir effect is the Van-der-Waals attraction between two plates separated by a distance large enough so that retardation effects are important. In Dirac gauge, you have electrostatic forces and photons, and the electrostatic force is not important, and the photon vacuum energy reduction is just due to the different photon modes that are allowed at different distances.
This is not mysterious, and it shouldn't be presented as mysterious. But it is a source of endless pseudoscience about machines that extract vacuum energy. The Casimir force doesn't allow you to extract any more energy than any other attractive effective interaction.