  We study here an alternative technique to probe the Dark Ages (DA) and the
Epoch of Reonization (EoR) that makes use of the Comptonization of the CMB
spectrum modified by physical effects occurring during this epoch related to
the emergence of the 21-cm radiation background. Inverse Compton scattering of
21-cm photon background by thermal and non-thermal electrons residing in the
atmospheres of cosmic structures like galaxy clusters, radiogalaxy lobes and
galaxy halos, produces a specific form of Sunyaev-Zel'dovich effect (SZE) that
we refer to as SZE-21cm. We derive the SZE-21cm in a general relativistic
approach which is required to describe the correct spectral features of this
astrophysical effect. We calculate the spectral features of the thermal and
non-thermal SZE-21cm in galaxy clusters and in radiogalaxy lobes, and their
dependence on the history of physical mechanisms occurring during the DA and
EoR. We study how the spectral shape of the SZE-21cm can be used to establish
the global features in the mean 21-cm spectrum generated during and prior to
the EoR, and how it depends on the properties of the (thermal and non-thermal)
plasma in cosmic structures. We find that the thermal and non-thermal SZE-21cm
have peculiar spectral shapes that allow to investigate the physics and history
of the EoR and DA. Its spectrum depends on the gas temperature (for the thermal
SZE-21cm) and on the electrons minimum momentum (for the non-thermal SZE-21cm).
The global SZE-21cm signal can be detected (in $\sim 1000$ hrs) by SKA1-low in
the frequency range $\nu \simgt 75-90$ MHz, for clusters in the temperature
range 5 to 20 keV, and the difference between the SZE-21cm and the standard SZE
can be detected by SKA1 or SKA2 at frequencies depending on the background
model and the cluster temperature. [abridged]
