We study the electric-field-induced birefringence and orientational order in the isotropic phase of aqueous suspensions of exfoliated natural beidellite clay particles, thin (L = 0.65 nm) flat charged sheets with high aspect ratio, D/L ≈ 300. Our electric birefringence experiment is optimized for aqueous suspensions of colloidal particles, with a high frequency a.c. electric field, ν ≈ 1 MHz, applied by two external electrodes to a thin flat sample, sealed in an optical capillary. In isotropic and biphasic samples, we observed strong field-induced birefringence Δn(E), saturating at moderate E(sat) field to a plateau Δn(sat) proportional to the volume fraction ϕ. The field-induced order parameter S(E) is negative and saturates to S(sat) = -0.5 above E(sat). This corresponds to a perfect "anti-nematic" order, i.e. the normals of the beidellite particles are perpendicular to the field, without any preferred azimuthal direction. The measured specific excess polarizability ΔA(sp) is among the highest data reported for other strongly anisometric dielectric and metal particles. We explain the high ΔA(sp) value with the strong induced polarization of the electric double layer of counterions at the charged particle/electrolyte interface. The estimated equivalent conductivity of the beidellite particle K(eq) = 2 K(σ)/L is several orders of magnitude larger than the bulk conductivity of the electrolyte K(e), resulting in a metal-like behavior of the beidellite disks under field. In the isotropic regions of biphasic nematic/isotropic samples, the excess polarizability is further enhanced by an order of magnitude, indicating collective reorientation of the particles. We propose that this enhancement might be due to pretransitional fluctuations of the spontaneous nematic order S(N) of the colloidal suspension and/or formation of chains of particles, with antinematic order of the beidellite disks in the chains.