I literally asked Gemini this question yesterday:

Why does The Dude hate the fucking Eagles?

In The Big Lebowski, The Dude’s legendary hatred for the Eagles isn't just a random joke—it’s a carefully crafted character detail that highlights the cultural divide between the 1960s counterculture and the 1970s "Me Decade."
Here is why the "Duderino" can't stand them:
1. Cultural Identity (Hippies vs. "Rich Hippies")
The Dude is a relic of the 1960s—a former member of the "Seattle Seven" and a draft dodger who values authenticity and gritty, anti-establishment vibes (represented by his love for Creedence Clearwater Revival).
• The Gritty: Creedence felt like swampy, blue-collar rock.
• The Slick: To the Dude, the Eagles represented the "corporatization" of the hippie movement. They were seen as a highly polished, commercially successful "soft rock" machine that took the edge off the counterculture and turned it into a smooth, radio-friendly product. 
2. The "Lazy" Vibe
In the famous taxi scene, the song playing is "Peaceful Easy Feeling." For a guy like the Dude, who is currently being harassed by nihilists, a corrupt police chief, and a millionaire namesake, the song's breezy, "everything is fine" California sentiment feels patronizing.
3. Behind the Scenes: T-Bone Burnett
The film’s music consultant, T-Bone Burnett, actually hated the Eagles in real life. He was the one who suggested the line to the Coen Brothers. Burnett famously remarked that the Eagles "single-handedly destroyed that whole scene that was brewing back then," referring to the more authentic country-rock movement of the early '70s. 
4. The "Hotel California" Irony
The movie features a Spanish flamenco cover of "Hotel California" by the Gipsy Kings during the introduction of Jesus Quintana. This serves as a final insult: the only time we hear an Eagles song in a "cool" or stylized way, it’s a cover performed by a rival, effectively stripping the original band of their own work within the Dude's world. 
Fun Fact: The line actually helped the production budget. The Coen Brothers wanted to use a Rolling Stones song ("Dead Flowers") for the credits, but the rights owner, Allen Klein, wanted $150,000. When Klein saw a rough cut of the movie and heard the Dude say, "I hate the fucking Eagles, man!" he reportedly stood up and said, "That’s it, you can have the song!" just because he hated the Eagles, too.