Green Day isn't bad. Well, let me rephrase, Green Day never used to be bad. I was a big fan of them back in the day (Kerplunk, Dookie, Insomniac - all decent albums). Unfortunately, their music grew stale, and lacks that appeal they had in the 90's. "American Idiot" definitely blows.
Same with Offspring, was a big fan of them as well. Ignition (1992) and Smash (1994) were in my CD player alot, but after Ixnay on the Hombre they really started to suck. I pretty much gave up on them after, "Pretty Fly For a White Guy"... Wouldn't hold it against someone for liking them back in the day though.
Personally, although being decent bands in that same-ish time period, I don't consider either of them "alternative rock". More of an offshoot of punk and ska kind of stuff.