No and yes.
First, there is no such thing as negative mass. Antiparticles have exactly the same mass as their matter counterparts. (E=mc², remember, so the fact that matter and antimatter release nonzero energy when they annihilate tells you that their masses can't cancel.)
Now, is there an anti-Higgs? Yes, and it's the Higgs boson! The fundamental bosons--such as the photon--are "self-conjugate", meaning that they are their own anti-particles. [Geek alert: some bosons, such as the W, carry charge, and are not exactly self-conjugate, but both of the charge states are of equal rank: the W+ and the W- have equal claims on being the "matter" particle.]
Personally, I don't think of these "gauge bosons" as matter at all, never mind the fact that some of them are quite massive! I think of them as particles of force. The universe is composed of matter and force, and both are quantized into particles.
In the religious spirit of the threads title, I submit that quarks and leptons are the "nouns" of God's Word, while the gauge bosons are the verbs.