I dont know this system personally but I have some experience with water technology from an industrial standpoint. Halo 5 apparently uses magnetic ion system as their way to address the magnesium and calcium. While salt based water softeners actually ion exchange and REMOVE the Ca and Mg ions, some other systems try to use technology to NEUTRALIZE the ions. For the systems that neutralize, the tests will still show hardness because the elements are still there, but the idea is that they will not scale and be able to pass through without depositing on fixtures and water heating elements.
Of all the non-salt based systems, TAC or template assisted crystallization is the most effective and proven in independent testing to reduce scale by >90%. A lot of industrial systems use TAC. This Halo magnetic system on the other is probably the most controversial because study results are mixed. Best case scenario it may seemingly reduce scale by 50%, but I'm not optimistic. I doubt it would be worth it for you.
For eczema, its hard to say what would be most beneficial. A lot is trial and error because every case is a bit different, but if you want to actually remove Ca and Mg ions then salt system is the only method.