There is room in various models for particles with 'negative mass', others rule them out, no proof of them currently exists either way. Regardless the particles involved in Hawking radiation, typically photons, are massless particles, photons, you generally assume virtual particles are generated very close to each other and for Hawking Radiation they are essentially assumed to be the absurdly tiny fraction of a distance one either side of an event horizon, one inside that prison, the other just a near non-existent fraction of a distance outside where the escape velocity is 99.999999% of light speed, so only those light speed particles can escape. The black hole loses mass because while a photon has no 'rest mass' it still has energy and E=mc² applies, giving the photon an effective mass that the black hole has to lose, nothing weird there, the stars and us all lose mass this way, it's just absurdly tiny... well not really tiny for a star, the sun loses about 10,000 tons of mass per second to raw photon emission, you or I, emitting around a few watts of infrared radiation, lose like a picogram a second or something, admittedly many billions of proton masses per second, but in both cases virtually nothing relative to the object's mass. Bigger and hotter objects lose at a greater rate, even, and especially, if their only type of emission is blackbody radiation.

It's also probably worth mentioning that the classic example of HR is a little watered down for public consumption, there's more going on then that. Specifically that the particles are essentially black body radiation emerging counter-intuitively inverse to mass. Small black holes die far faster, bigger ones not at all, not yet, because they absorb more even in an absolute vacuum from cosmic microwave background radiation then they emit. Still there are few things bigger, in terms of mass, or hotter, than a black hole, so I've always found it ironic that they emit less energy as they get bigger... especially considering they have far more actual area/volume for those virtual particle losses to occur in as they get bigger. Double a black hole's mass and you double it's radius, increasing it's event horizon's surface by a factor of 4, Triple and you triple radius, event horizon rises by 9x, etc. So in the classic visual example one should expect Hawking Radiation to scale by rise as the square of mass, which it doesn't, because the classic HR example is basically BS and really has a fairly limited relation to reality. One of the reasons I don't really like to talk about it or black holes in general, I hate the examples that get used, many are sensationalized and many others are kinda-bullshit to avoid going into the math.