Yes, this is basically the problem, along with his clarification "By the phrase "exotic matter" I mean matter that has negative energy and therefore anti-gravitates, i.e. repels."
First of all, the Casimir force is attractive, not repulsive. The plates get pushed together. It also has nothing to do with gravity - the derivation of the force relies on the electric and magnetic field boundary conditions applied to the conducting plates; it's purely an electromagnetic phenomenon. You eliminate photon standing wave modes (of one polarization) because the field needs to go to zero at the boundary. The QM solution says for the vacuum, each state has an energy of hv, so there are fewer photon modes inside the plates than outside. The energy density imbalance means there is a pressure that pushes them together. In short, if the plates are a micron apart, then no photon states of 2 microns or longer can exist in between the plates (plus more that wouldn't form standing waves) but they do exist outside the plates.
There's no gravity involved, and no matter identified with it, AFAICT.
Maybe his point is that if the Casimir force exists maybe there's something like this for gravity, but still, the Casimir force isn't tied to exotic matter and it's attractive and it relies on there being boundary conditions for E and B fields that AFAIK don't exist for gravity, so it would be a really weak analogue for the kind of exotic matter you need for wormholes to become stable