Despite being largely limited to bulk phenomena, well-known theoretical models of superconductivity like the Bardeen–Cooper–Schrieffer and Ginzburg–Landau theories have played a key role in the development of superconducting quantum devices. In this letter, we present a hydrodynamic non-relativistic scalar electrodynamic theory capable of describing systems comprising superconducting materials of arbitrary shape and apply it to predict the existence of a negative (attractive) pressure between planar superconducting bodies. For conventional superconductors with London penetration depth λL ≈ 100 nm, the pressure reaches tens of N/mm2 at angstrom separations.