What does the phrase “Due to Lorentz invariance, only the Higgs particle can have a non-zero expected value in a vacuum” mean?
It is the Higgs field, not the Higgs particle, that has a nonzero vacuum expectation value (VEV).
The Higgs field is a scalar field. Scalar fields don’t have a direction in spacetime; they just have a value. Other fields such as spinor fields (for electrons, quarks, and neutrinos) and vector fields (for photons, gluons, and weak bosons) do have a direction in spacetime. (In the case of spinors, I’m simplifying a bit.) Lorentz transformations change this direction, so a VEV for spinor and vector fields could not be Lorentz-invariant; for the scalar Higgs it can.