We investigated optical properties of nanostructured metal-dielectric multilayered lattices under the conditions of epsilon-near-zero (ENZ), a concept derived from the effective-medium approach (EMA). We theoretically found that the periodic array of metallic nanolayers may exhibit either superlensing driven by broadband canalization from point emitters or single-polarization double refraction, and conventional positive as well as negative, even at subwavelength regimes. For the latter case, we formulated a modified EMA, and subsequently a generalized refraction law, that describes both refractive behaviors concurrently. The modal coupling of plasmonic lattice resonances, and nonlocality induced by partial screening across the nanolayer length, are responsible for these distinct effects. Numerical simulations show that deep-subwavelength lensing along the optical axis of Ag-GaAs metamaterial is clearly enhanced at optical wavelengths. On the other hand, transverse-magnetic-polarized radiation that is obliquely incident on the ENZ periodic nanostructures with the same materials in the infrared (around ) undergoes double refraction neighboring beamsplitting.