Quasinormal modes of black holes were previously calculated in a non-linear electrodynamics and in the Gauss-Bonnet gravity theory. Here we take into consideration both of the above factors and find quasinormal modes of a (massive) scalar field in the background of a black hole in the five-dimensional Einstein-Gauss-Bonnet gravity coupled to a non-linear electrodynamics having Maxwellian weak-field limit. For the non-linear electrodynamics we considered the high frequency (eikonal) regime of oscillations analytically, while for the lower multipoles the higher order WKB analysis with the help of Pade approximants and the time domain integration were used. We found that perturbations of a test scalar field violate the inequality between the damping rate of the least damped mode and the Hawking temperature, known as the Hod's proposal. This does not exclude the situation in which gravitational spectrum may restore the Hod's inequality, so that only the analysis of the full spectrum, including gravitational perturbations, will show if the quasinormal modes we found here for the scalar field can be a counterexample to the Hod's conjecture or not. We also revealed that in such a system, which includes the higher curvature corrections and non-linear electrodynamics, for perturbations of a massive scalar field there exists the phenomenon of the arbitrary long lived quasinormal modes - quasiresonances.