Note that the Higgs Boson has no part in the mass (and presumably gravitational interaction) of elementary fermions like quarks or electrons, and even less to do with the mass of hadrons (whose mass is the mass of quarks + the mass of the extraordinary amount of energy sticking the quarks together, which dwarfs the mass of the quarks by ~100:1). Even if all elementary particles were massless, protons, nuclei, and atoms would still have most of the mass they have today (though of course many other interactions would be very different indeed).