Polaritons in layered materials-including van der Waals materials-exhibit hyperbolic dispersion and strong field confinement, which makes them highly attractive for applications including optical nanofocusing, sensing and control of spontaneous emission. Here we report a near-field study of polaritonic Fabry-Perot resonances in linear antennas made of a hyperbolic material. Specifically, we study hyperbolic phonon-polaritons in rectangular waveguide antennas made of hexagonal boron nitride (h-BN, a prototypical van der Waals crystal). Infrared nanospectroscopy and nanoimaging experiments reveal sharp resonances with large quality factors around 100, exhibiting atypical modal near-field patterns that have no analogue in conventional linear antennas. By performing a detailed mode analysis, we can assign the antenna resonances to a single waveguide mode originating from the hybridization of hyperbolic surface phonon-polaritons (Dyakonov polaritons) that propagate along the edges of the h-BN waveguide. Our work establishes the basis for the understanding and design of linear waveguides, resonators, sensors and metasurface elements based on hyperbolic materials and metamaterials.