We have defined a sample of 63 active galactic nuclei with strong forbidden high-ionization line (FHIL) emission. These lines, with ionization potentials ≳100 eV, respond to a portion of the spectrum that is often difficult to observe directly, thereby providing constraints on the extreme ultraviolet-soft X-ray continuum. The sources are selected from the Sloan Digital Sky Survey (SDSS) on the basis of their [Fe x]λ6374 Å emission, yielding one of the largest and the most homogeneous samples of FHIL-emitting galaxies. We fit a sequence of models to both FHILs ([Fe xi], [Fe x] and [Fe vii]) and lower ionization emission lines ([O iii], [O i], Hα, [N ii], [S ii]) in the SDSS spectra. These data are combined with X-ray measurements from the ROSAT satellite, which are available for half of the sample. The correlations between these parameters are discussed for both the overall sample and subsets defined by spectroscopic classifications. The primary results are evidence that (1) the [Fe x] and [Fe xi] lines are photoionized and their strength is proportional to the continuum flux around 250 eV, (2) the FHIL-emitting clouds form a stratified outflow in which the [Fe x] and [Fe xi] source regions extend sufficiently close to the broad-line region that they are partially obscured in Seyfert 2s, whereas the [Fe vii] source region is more extended and is unaffected by obscuration, (3) narrow-lined Seyfert 1s (NLS1s) tend to have the strongest [Fe x] flux (relative to lower ionization lines) and (4) the most extreme [Fe x] ratios (such as [Fe x]/[O iii] or [Fe x]/[Fe vii]) are found in the NLS1s with the narrowest broad lines and appear to be an optical-band indication of objects with strong X-ray soft excesses.