Negative mass is something quite different, and has been hypothesised as being required for wormhole travel, as in the centre repulsion, rather than attraction, is required, and the only way to acquire this from gravitation forces is with negative mass. This then implies (or is implied by, I forgot the reasoning) negative energy.
Of course, the wormhole thing is entirely incidental, but then again I haven't seen anything on this for a few years so god knows what's happening now.
I'm afraid it would; special relativity has nothing to do with the phases of matter, merely mass and relative velocity.