Phys. Rev. D 105, 084020 (2022) In this paper we study properties that the vacuum must possess in the minimal extension to the teleparallel equivalent of general relativity (TEGR) where the action is supplemented with a quadratic torsion term. No assumption is made about the weakness of the quadratic term although in the weak-field regime the validity of our previously derived perturbative solution is confirmed. Regarding the exact nature of the vacuum, it is found that if the center of symmetry is to be regular, the mathematical conditions on the tetrad at the isotropy point mimic those of general relativity. With respect to horizons it is found that, under very mild assumptions, a smooth horizon cannot exist unless the quadratic torsion coupling, $\alpha$, vanishes, which is the TEGR limit (with the Schwarzschild tetrad as its solution). This analysis is then supplemented with computational work utilizing asymptotically Schwarzschild boundary data. It is verified that in no case studied does a smooth horizon form. For $\alpha > 0$ naked singularities occur which break down the equations of motion before a horizon can form. For $\alpha < 0$ there is a limited range of $\alpha$ where a vacuum horizon might exist but, if present, the horizon is singular. Therefore physically acceptable black hole horizons are problematic in the studied theory at least within the realm of vacuum static spherical symmetry. These results also imply that static spherical matter distributions generally must have extra restrictions on their spatial extent and stress-energy bounds so as to render the vacuum solution invalid in the singular region and make the solutions finite.