As I understand it, the main thing pushing the rest of the star from falling in would be the momentum of the light emitted by the accretion disk. Those might be early-population stars made of light elements, but a lot of the radiation pressure would just go right through the layer of star material, or even worse, be refracted such that there's also radiation pressure going inward.
The only other accelerations that might be acting outward would be matter that escapes from the accretion disk, which would be concentrated around the equator and result the quasi-star just turning into an accretion disk, and the spin of the star about its axis, which would allow the poles to collapse and therefore turn the quasi-star into an accretion disk.
There is no stable arrangement in which a quasi-star can exist. Maybe that's why they give it a lifetime of a million years, less than a hair's width on the cosmic scale of time. I wouldn't glorify that with a name; a hollow celestial body is just what happens when a black hole winds up in its center. It wouldn't be worth mentioning to anyone who specializes in astronomy as anything except a discrete part of a sufficiently massive supernova's life cycle.