I have every American Rifleman magazine printed since 1928, and have nearly completed a directory to their information within. Plus, a number of books, new and old.

Numerous references state that the nominal bore diameter of the 28-gauge is .550 inch.
This makes sense, because I knew a guy in the early 1980s who used .54-caliber muzzleloader bullets in his 28-gauge for slugs. I think he used the classic Thompson Center Maxi-Hunter lead bullet.
I would think one could use .53-caliber lead balls inside a shot cup as well. Might be a fun experiment. The smaller ball is needed, to fit inside the shot cup, which is probably .54 caliber or so.
Dixie Gun Works sells hand-cast pure lead balls at a reasonable price. Speer or Hornady make .53-caliber swaged balls.

I couldn't find any references to loading buckshot or slugs for the 28-gauge, even in my older Lyman reloading books and back issues of Handloader and Rifle magazines.
I'm still in the process of adding some references to my directory, so there may be information I have not yet found.

In the August 1931 issue of the American Rifleman, there is, "The 28-Bore Shotgun" by Charles Askin, on page 9.
Interestingly he writes, "There should have been plenty of demand for the 28-bore this last twenty years, but oddly enough there wasn't, they tell me."
He laments that the niche of a small-bore, lightweight shotgun has been taken by the .410, adding that he could never shoot it "worth a whoop" because it is too short-barreled, light and has untypical stock dimensions.
Askins does not mention slug or buckshot loads.

I don't know of anyone making slug or buckshot loads for the 28 gauge. I think you'd have to make it yourself, or contact a company that specializes in odd, obsolete or custom ammo.

Two good specialty companies that might have 28 gauge slug or buckshot loads are Gad Custom Cartridges and Quality Cartridge. An internet search will find them.

Good luck. Let us know what you find out.