Out of universe, this is probably T Bone Burnett talking. From IMDb:
T Bone Burnett acted as music consultant for the movie, and helped Joel Coen and Ethan Coen establish the Dude's taste in music. Burnett selected many of the existing songs in the movie, and also suggested the Dude's hatred towards The Eagles (Burnett himself is not a fan either). One of the band's member, Glenn Frey, was reportedly so dismayed about this that he once even angrily confronted Jeff Bridges when they met at a party.
"[The Eagles] sort of single-handedly destroyed that whole scene that was brewing back then."
On a side note, the Dude's aversion to the Eagles also helped the team to secure a desired song, saving them as much as $150,000. From Wiki:
[Burnett] had a tough time securing the rights to Townes Van Zandt's cover of the Rolling Stones' "Dead Flowers", which plays over the film's closing credits. Former Stones manager Allen Klein owned the rights to the song and wanted $150,000 for it. Burnett convinced Klein to watch an early cut of the film and remembers, "It got to the part where the Dude says, 'I hate the f@$&in' Eagles, man!' Klein stands up and says, 'That's it, you can have the song!' That was beautiful."
It's possible a hardcore, counter-culture 60s hippie like the Dude would consider the Eagles sellouts. From The Big Lebowski: An Illustrated, Annotated History of the Greatest Cult Film of All Time:
Even the choice of Creedence represents the more rebellious side of classic rock. [...] Creedence symbolizes the polar opposite of the band the Dude openly hates, the Eagles. In the mid-1990s, the Eagles became the first mainstream rock band to sell tickets for more than $100. If Creedence had a dash of antiwar insurgency, the Eagles were all Baby Boomer contentment, a feeling encapsulated in the song "Peaceful Easy Feeling". [...]
Eagles fans should forgive the Dude for reacting so harshly to [that song]. It was, after all, the Dude's perceived lackadaisical hippie way of life that Jeffrey Lebowski attempted to exploit, and the lazy-day vibe of the Eagles' "Peaceful Easy Feeling" [...] is hippiedom as filtered by Boomer nostalgia. By this time in the film, it's been made clear that there's far more to the Dude than any peace-and-love cliche.