In the very early universe, there were no particles, only photons zipping about at immense speeds. Those photons must have collided with each other at some point or other, resulting in one of them absorbing the other and therefore creating slightly higher energy densities in certain areas (that is, quantum fluctuations in the early universe). Then energy cooled and the photon concentrations turned into tiny concentrations of mass (which particles later began to form atoms as the universe cooled). As the cosmos expanded, the mass concentrations began to form nebulae as they were spread out over the increasing distances -- and gravity began to draw more matter to them and shrink them into the first stars.
How Particles Acquire Mass
The mass-energy equivalence states that as a photon is emitted from an electron or any other particle, the mass of the particle increases, because the energy (e) is equal to the mass (m) times the speed of light squared (c). This is because the net mass (the energy + mass) needs to stay the same to prevent a particle from changing, say, from an electron to a quark and so forth. This creates a logical explanation for how mass is acquired, and the Higgs mechanism is not required.