In recent years, the large electric field enhancement and tight spatial confinement supported by the so-called epsilon near-zero (ENZ) mode has attracted significant attention for the realization of efficient nonlinear optical devices. Here, we experimentally demonstrate ENZ photonic gap antennas (PGAs), which consist of a dielectric pillar within which a thin slab of indium tin oxide (ITO) material is embedded. In ENZ PGAs, hybrid dielectric-ENZ modes emerge from strong coupling between the dielectric antenna modes and the ENZ bulk plasmon resonance. These hybrid modes efficiently couple to free space and allow for large enhancements of the incident electric field over nearly an octave bandwidth, without the stringent lateral nanofabrication requirements of conventional plasmonic or dielectric nanoantennas. To understand the modal features, we probe the linear response of single ENZ PGAs with dark field scattering and interpret the results in terms of a simple coupled oscillator framework. Third harmonic generation (THG) is used to probe the ITO local fields and large enhancements are observed in the THG efficiency over a broad spectral range. Surprisingly, sharp peaks emerge on top of the nonlinear response, which were not predicted by full wave calculations. These peaks are attributed to the ENZ material's nonlocal response, which once included using a hydrodynamic model for the ITO permittivity improves the agreement of our calculations for both the linear and nonlinear response. This proof of concept demonstrates the potential of ENZ PGAs, which we have previously shown can support electric field enhancements of up to 100--200X, and the importance of including nonlocal effects when describing the response of thin ENZ layers. Importantly, inclusion of the ITO nonlocality leads to increases in the predicted field enhancement, as compared to the local calculation.