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As part of an ongoing study of the neutral hydrogen (H I) morphology and kinematics of "shell" elliptical galaxies, we present Very Large Array observations of two shell galaxies with aligned shells, Arp 230 and MCG -5-7-1. Our data provide the first H I images of Arp 230 and deeper images of MCG -5-7-1 than previously reported. Optical images of Arp 230 reveal a bright, aligned, interleaved shell system, making it an ideal candidate for "phase-wrapped" shell formation following a radial encounter with a smaller companion. The fainter, non-interleaved shells of MCG -5-7-1 do not clearly favor a particular formation scenario. The H I we detect in both galaxies extends to nearly the same projected distance as the optical shells. In Arp 230 this gas appears to be anti-correlated with the aligned shells, consistent with our expectations for phase-wrapped shells produced in a radial encounter. In MCG -5-7-1, we observe gas associated with the shells making a "spatial wrapping" or looping scenario more plausible. Although the extended gas component in both galaxies is unevenly distributed, the gas kinematics are surprisingly regular, looking almost like complete disks in rotation. We use the H I kinematics and optical data to determine mass-to-light ratios M/L_B of 2.4^+3.0_–0.5(at 13.5 kpc, 4.5 R_e) for Arp 230 and M/L_B of 30±7 (at 40 kpc, 7 R_e) in MCG -5-7-1. In both systems we find that this ratio changes as a function of radius, indicating the presence of a dark halo. By comparing orbital and precession timescales, we conclude that the potentials are slightly flattened. We infer a 5%-10% flattening for Arp 230 and less flattening in the case of MCG -5-7-1. Finally, we present images of the H I associated with the inner disk or (polar) ring of each galaxy and discuss possible explanations for their different present-day star formation rates. We detect total H I masses of 1.1x10⁹ M_☉ in Arp 230 and 3.7x10⁹ M_☉in MCG -5-7-1. Both systems have H I masses, scale sizes, and regular kinematics similar to those of non-aligned shell galaxies we have previously studied (Cen A and NGC 2865). Furthermore, we (re-)emphasize in this paper that shell galaxies such as MCG -5-7-1, along with previously studied galaxies NGC 5128 (Cen A) and NGC 2865, are unique in that they provide evidence of recent accretion with gas and collisionless stars showing clear association, though the displacement suggests the presence of significant gas-dynamical interaction.