Quasiparticle tunneling measurements of the high-temperature superconductors HgBa 2 Ca nϪ1 Cu n O 2nϩ2ϩ␦ (Hg-12(nϪ1)n,nϭ1,2,3) are considered in the context of d x 2 Ϫy 2 symmetry of the superconducting order parameter and a two-dimensional ͑2D͒ van Hove singularity ͑vHs͒ related to saddle points in the electronic band structure. Normal-metal-insulator-superconductor tunneling spectra taken at 4.2 K with a scanning tunneling microscope on Hg-1212 c-axis epitaxial films, as well as on Hg-1201 and Hg-1223 polycrystalline samples, show distinct gap characteristics which cannot be easily reconciled with the simple s-wave BCS density of states. The data are analyzed with the nodal d-wave gap function ⌬ k ϭ⌬ 0 (cos k x Ϫcos k y )/2 and the 2D tight-binding electronic dispersion k ϭϪ2t(cos k x ϩcos k y )ϩ4tЈ(cos k x cos k y )Ϫ, using the quasiparticle tunneling formalism for elastic and specular transmission. The analysis indicates a highly directional and energy-dependent spectral weighting, related to the gap anisotropy and band-structure dependence of the tunneling matrix element ͉T͉ 2 , and successfully explains the observed gap spectra. Values for the d-wave gap maximum are determined to be ⌬ 0 Ϸ33, 50, and 75 meV, respectively, for optimally doped Hg-1201, Hg-1212, and Hg-1223, corresponding to reduced-gap ratios of 2⌬ 0 /k B T c Ϸ7.9, 9.5, and 13. These ratios are substantially larger than the BCS weak-coupling limit of 3.54. A comparison with data from other high-T c cuprates indicates an overall trend of 2⌬ 0 /k B T c rising with T c , in violation of BCS universality.