Boiling chips for concentrating sulfuric acid
Clean porcelain or glass shouldn't react with concentrated sulfuric acid.
For boiling chips you want to use homogeneous material with a substantial/developed surface:

Porous unglazed ceramic.
Raschig rings, which are also widely used industrially for the similar distillation purposes, are probably the ideal choice.
An alternative for a consumer market would be porous ceramic rings for aquarium filters, which is basically the same thing (stick with the white ones).

The majority of glass types (borosilicate glass, quartz, …).
Broken pieces of chemical glassware or light bulbs would do.
For higher efficiency, use glass capillaries or sintered glass pieces.


Potential cause of cloudy solution when you used mug pieces might be the glaze on top of porcelain or the porcelain itself if it hasn't been properly thermally treated (remaining binder), and, of course, porcelain dust particles leftovers may float around forming a suspension.
Yellow color of sulfuric acid when you used glass pieces is likely an indicator that acid came in contact with organics (e.g. oil, grease).
Pure colorless glass won't cause this coloration.
On a side note, there is no need to use grease at all as sulfuric acid itself is quite viscous and can serve as a sealant for the joints (if you are using any).
Notes, advises

If the liquid has been cooled down, make sure to add at least one new chip before heating again to the boiling point (prevents superheating).
Avoid using boiling chips for vacuum distillation.
Instead, use mechanical stirring or bubble inert gas (argon, nitrogen) through a submerged glass capillary.
Boiling chips don't work under vacuum.
Never add boiling chips to a hot solution as it will result in rapid vaporization and the liquid is going to boil over or bump.
Keeping a substantial amount of sodium carbonate nearby might come in handy in neutralizing spilled acid, if there is any.