I've never heard anyone argue that it would react any differently to gravity than normal matter. After all, the only difference is the electrical charge of the particles. They have the same mass.
Of course, regular old hydrogen "falls up", being lighter than air. :) I guess if gravity is truly reversed for antihydrogen, it would fall up even faster. But I highly doubt it's actually gravitationally repulsive.
Or perhaps must be affected by antigravity the way matter is by gravity. Assuming antigravity exists. Whether that's correct at all I don't know, but that's the logic being used.
So if it "reacts the opposite" to gravity, what does that mean?
I would see no reason for Newton's law to not hold true around our scale, so repelling means F=Gm1m2/r^2 which only makes sense if one of the masses is actually negative. Just the thought of it looks like fun.