USAA has been better than anyone else I've used as far as claims processes go. I've had to make comprehensive claims with USAA and they have been seamless.
I had to switch to esurance earlier this year after a PCS, since I saved almost $600 every 6 months. I've already had to make two claims, and it has just been outrageous. Additionally, I've found that I can't get the same coverage levels that I had with USAA. For example, my liability was $500,000 per accident with USAA. I've had trouble finding anything over $300,000 with other companies. Also, I always ran $0 deductibles with USAA, I haven't been able to replicate that since switching.
Despite the savings I'm getting this year, I think I may switch back next year even if the premium is slightly higher.
I can't speak for homeowners insurance claims other than when my car got broken into and items were pilfered, and when I made a claim for a broken laptop and ipod touch screen. Both were paid immediately with no hassle and money issued for a replacement.