We propose a new platform to detect signatures of the presence of Majorana bound states (MBSs) in the Fraunhofer pattern of Josephson junctions featuring quantum spin Hall edge states on the normal part and Majorana bound states at the NS interfaces. We use a tight-binding model to demonstrate a drastic change in the periodicity of the Fraunhofer pattern when comparing trivial and nontrivial regimes. We explain these results in terms of the presence of additional parallel-spin electron-hole reflections, which due to the spin-momentum locking, occur as cross Andreev reflections, accumulating a different magnetic flux and yielding a change in the Fraunhofer periodicity. We show that this detection scheme exhibits some advantages compared to previous ones as it is robust against disorder and finite temperature and works in equilibrium. Furthermore, we introduce a scattering model that captures the main results of the microscopic calculations with MBSs and extend our discussion to the main differences found using accidental zero-energy Andreev bound states.