  We demonstrate that in photonic gap antennas composed of an epsilon-near-zero
(ENZ) layer embedded within a high-index dielectric, hybrid modes emerge from
the strong coupling between the ENZ thin film and the photonic modes of the
dielectric antenna. These hybrid modes show giant electric field enhancements,
large enhancements of the far-field spontaneous emission rate and a
unidirectional radiation response. We analyze both parent and hybrid modes
using quasinormal mode theory and find that the hybridization can be well
understood using a coupled oscillator model. Under plane wave illumination,
hybrid ENZ antennas can concentrate light with an electric field amplitude
$\sim$100 times higher than that of the incident wave, which places them on par
with the best plasmonic antennas. In addition, the far-field spontaneous
emission rate of a dipole embedded at the antenna hotspot reaches up to
$\sim$2300 that in free space, with nearly perfect unidirectional emission.
