Does the existence of Higgs boson forbid the possibility of negative mass? Since negative mass seems not to be impossible according to relativity, does the existence of a fixed particle responsible of mass contradicts with it? 
 A: I don't think it is reasonable to argue that specifically the Higgs mechanism forbids negative masses because QFT as a whole forbids negative masses. It's hard to see how any quantum field theory could be constructed with negative masses and remain consistent.
In fact the Higgs mechanism tells us that the particle mass is given by:
$$ m = \sqrt{yH_0} $$
where $y$ is the coupling to the Higgs field and $H_0$ is the Higgs VEV. There are of course positive and negative roots, but we only take the positive root because, well, taking the negative root would cause all sorts of mayhem that isn't experimentally observed.
Negative mass is possible in general relativity because in GR mass is an unexplained quantity that can be given any value positive or negative. What QFT brings to the table is an attempt to explain exactly what mass is and how it is generated. I feel fairly safe in saying QFT doesn't allow negative masses, but whether that means negative masses are impossible is another question. That is, there could be some other mechanism for generating mass that we don't know about.
