This paper considers the capacity of sub-sampled analog channels when the sampler is designed to operate independent of the instantaneous channel realization, and investigates sampling methods that minimize the worst-case (minimax) sampled capacity loss due to channel-independent (universal) sampling design. Specifically, a compound multiband channel with unknown subband occupancy is considered, when perfect channel side information is available to both the receiver and the transmitter. We restrict our attention to a general class of periodic sub-Nyquist samplers, which subsumes as special cases sampling with modulation and filter banks. Our results demonstrate that under both Landau-rate and super-Landau-rate sampling, the minimax sampled capacity loss due to universal design depends only on the band sparsity ratio and the undersampling factor, modulo a residual term that vanishes at high signal-to-noise ratio. We quantify the capacity loss under sampling with periodic modulation and low-pass filters, when the Fourier coefficients of the modulation waveforms are randomly generated (called random sampling). Our results highlight the power of random sampling methods, which achieve minimax sampled capacity loss uniformly across all channel realizations and are thus optimal in a universal design sense.