Using boundary-integral projections for time-harmonic electromagnetic (EM) fields, and their numerical implementation, we analyze EM resonance in slabs of two-phase dielectric photonic crystal materials. We characterize resonant frequencies by a complex Floquet--Bloch dispersion relation $\omega = W(\beta)$ defined by the existence of a nontrivial nullspace of a pair of boundary-integral projections parameterized by the wave number $\beta$ and the time-frequency $\omega$. At resonant frequencies, the crystal slab supports a source-free EM field. We link complex resonant frequencies, where the imaginary part is small, to resonant scattering behavior of incident source fields at nearby real frequencies and anomalous transmission of energy through the slab. At a real resonant frequency, the source-free field supported by the slab is a bound state. We present numerical examples which demonstrate the effects of structural defects on the resonant properties of a crystal slab and surface waves supported by a dielectric defect.