So right now I have a female breloom with technician ability, and a male serperior. I also have given the breloom an everstone to pass down her nature.
Of the 15 shroomish I've hatched so far, none have had technician, but all have had the mother's nature.
Does the everstone prevent the technician ability from passing down?
You're just unlucky. If a female DW Pokémon is bred with a non-Ditto father, there should be a 60% chance that the DW ability will be passed down. If I recall, the Everstone passes down nature only.