The most important thing is hay, not the color. Green hay is sometimes dyed because buyers prefer green hay. But yellow hay might have been left out in the sun longer, and that means it has formed more vitamin D.
What is important though is variety. You can buy different hays and mix them or feed them in rotation, just so the bun gets hay that has grown on various soils under various conditions, harvested at various points in time (there are differences between 1st, 2nd and 3rd/4th cut).
You can use Timothy, oat, orchard, meadow + any other hay horses can eat like Bermuda grass, fescue, brome. You should avoid legume hays for adult buns (alfalfa, clover) but all the grassy hays are good. It should always smell good and be dry. Some growers declare nutritional content, because to horse owners, that’s important. It’s not unusual nowadays for horses to be on a hay-only diet, and rabbits and horses have very similar digestive systems and nutritional needs (adjusted for body weight, obviously).
As for your other bun, White discharge sounds a bit worrying. But if no vet has wanted to put him on antibiotics, you could try and de-dust the hay he gets. You can pour it out on a tarpaulin or a bed sheet and collect everything but the dusty short pieces back into a bag. Hay for pets should always be dust-free, but there are still differences from batch to batch, and not necessarily depending on the price tag.
Some buns are more sensitive to dustiness than others. A bun can even be allergic to certain hays. BB herself has a bun that’s tested allergic to timothy.