  We demonstrate that the James Webb Space Telescope (JWST) can detect infrared
(IR) excess from the blended light spectral energy distribution of spatially
unresolved terrestrial exoplanets orbiting nearby white dwarfs. We find that
JWST is capable of detecting warm (habitable-zone; T$_{\rm eq}$=287 K) Earths
or super-Earths and hot (400-1000 K) Mercury analogs in the blended light
spectrum around the nearest 15 isolated white dwarfs with 10 hrs of integration
per target using MIRI's Medium Resolution Spectrograph (MRS). Further, these
observations constrain the presence of a CO$_2$-dominated atmosphere on these
planets. The technique is nearly insensitive to system inclination, and thus
observation of even a small sample of white dwarfs could place strong limits on
the occurrence rates of warm terrestrial exoplanets around white dwarfs in the
solar neighborhood. We find that JWST can also detect exceptionally cold
(100-150 K) Jupiter-sized exoplanets via MIRI broadband imaging at $\lambda =
21\,\mathrm{\mu m}$ for the 34 nearest ($<13$ pc) solitary white dwarfs with 2
hrs of integration time per target. Using IR excess to detect thermal
variations with orbital phase or spectral absorption features within the
atmosphere, both of which are possible with long-baseline MRS observations,
would confirm candidates as actual exoplanets. Assuming an Earth-like
atmospheric composition, we find that the detection of the biosignature pair
O$_3$+CH$_4$ is possible for all habitable-zone Earths (within 6.5 pc; six
white dwarf systems) or super-Earths (within 10 pc; 17 systems) orbiting white
dwarfs with only 5-36 hrs of integration using MIRI's Low Resolution
Spectrometer (LRS).
