We address the problem of boundary s-wave superconductivity on rectangular honeycomb lattices: nanoflakes, armchair, and zigzag nanotubes. We discuss how the presence of edges and corners in these systems can significantly alter the superconducting correlations at a macroscopic length scale, leading to either nontrivial enhancement or suppression of the superconducting gap value near the boundaries. This in turn results in different critical temperatures of the gap closure at boundaries compared to the bulk gap. The effects are macroscopic but strongly depend on the atomic-level structure of the boundaries.