Well, a full grown American badger is approx. 90 cm long. assuming they are three times as long as they are tall, we can approximate a badger as a cylinder with volume of 0.063 meters^3. These are also very fat, very grumpy badgers of the Badgerius Cylindrius subspecies, which are known for, among other things, being perfectly cylindrical and incompressible.
The sun has a volume of 1.409×10^18 kilometers^3 , so it would take 2.252×10^28 badgers to fill it up. Alternatively written as 22,365,000,000,000,000,000,000,000,000 fully-grown badgers. Give or take. Therefore, 3.2 Quintillion badgers for every person on earth.
Furthermore, if you pile up that many badgers in one place, the ones on the inside are going to be compressed to an even smaller volume; the density of degenerate badgers could be very high. The density of the Sun is about 1.4 g/cm^3 and the density of a Sun-sized sphere of badgers could be higher.
Eurasian badgers weigh around 18 kg, which, assuming a density similar to that of water, makes for a total volume of about 0.018 meter^3. A badger with a volume of 0.063 meter^3 would weigh about 63 kilograms, or 139 pounds. So there’s room for a lot more badgers of this variety in the Sun than the previous calculations indicate.
African badgers are nonmigratory.
There would likely be fusion at the core, though probably not as strongly as in the Sun since badgers contain some hydrogen, but not in as high a proportion as the Sun. There would probably have to be more badgers added as the core collapses to maintain the size. Note, adding mushrooms and/or snakes probably won’t affect the physics much.
The real question is, if we used Honey Badgers, would they even care?