In 2016, the NATO Science for Peace and Security Programme funded research project ”Engineering Silicon Carbide for Border and Port Security” — E-SiCure was launched, its objective being the development of radiation-hard silicon carbide (SiC) based detectors of special nuclear materials (SNM), with the aim to enhance border and port security barriers. Detector prototypes based on SiC Schottky Barrier Diodes (SBDs) and neutron converter films were developed. This paper presents the results of a dedicated experimental testing campaign performed at the Jožef Stefan Institute (JSI) TRIGA reactor in which several SiC detector prototypes equipped with 10B and 6LiF converter films were irradiated in the Dry Chamber of the reactor. The obtained results demonstrate a clearly measurable neutron response, which varies linearly with the neutron flux. The measured particle spectra from the SiC detectors exhibit a clear structure, attributable to the nature and energy of secondary particles originating as reaction products from nuclear reactions involving 10B and 6Li isotopes. The determined sensitivity of the detectors, their active volume being 1 mm × 1 mm × 25 μm, 1 mm × 1 mm × 69 μm and 1 mm × 1 mm × 170 μm, was of the order of 2 × 10−5 counts per second, per unit of neutron flux counts s−1 per n cm−2s−1 (for neutron energies between 0 and 5 eV). Scaling the detection sensitivity by a factor of 105, i.e. to an array with a surface of around 20 cm × 2 m, comparable to large BF3 or 3He detectors, would theoretically enable an overall sensitivity of around 2 counts s−1 per n cm−2s−1, which is already comparable to typical neutron sensitivity values of gas detectors, in the range from several to over 100 counts s−1 per n cm−2s−1. Due to its outstanding tolerance to harsh environments (including high temperatures and radiation fields) and superior electronic properties when compared to other semiconductors, SiC is a promising base material for the fabrication of solid-state detectors with stable and long life-time. Improvements in sensitivity combined with the capability of fabricating large modules (SiC arrays), could make SiC an important detection technology, applicable also in the context of border and port security barrier monitoring.