Determining whether superconducting scanning tunneling spectroscopy (STS) is uniquely dictated by crystal structure constitutes a fundamental challenge in condensed matter physics. Here, we systematically investigate bulk FeSe single crystals, monolayer FeSe, and KCa₂Fe₄As₄F₂. We resolve one-, two-, and three-order checkerboard quantum-well structures that perfectly match the experimentally observed one, two, and three pairs of superconducting coherence peaks. In bulk FeSe, quantum wells promote real-space Cooper pairing and form degenerate antiferromagnetic checkerboard sublattices, yielding bosonic even-parity STS responses. In monolayer FeSe, mirror symmetry breaking suppresses Cooper pairing and induces nondegenerate ferromagnetic sublattice dichotomy, producing fermionic odd-parity STS spectra. We establish a universal gap scaling law Δ(T, ξ) = η(T)/ξ², where η(T) is a temperature-dependent prefactor and ξ denotes quantum-well depth that governs the number and magnitude of superconducting gaps. For KCa₂Fe₄As₄F₂, our predicted gap pairs of ±6.2 meV, ±5.6 meV, and ±4.2 meV are in excellent agreement with experimental results of ±6.2 meV, ±5.4 meV, and ±4.4 meV. This quantum-well mechanism unifies mirror symmetry breaking, checkerboard sublattice ordering, Cooper pairing, fermion-boson duality, and half-Bogoliubov states for STS interpretation, offering new insights toward a unified high-Tc superconductivity theory.