I know I'm 12 years late to comment on this, but I don't think Eddie hates Bon Jovi, just Jon's attitude and his "tricks". We all know what Jon is like.
A later date, in 2017 Pearl Jam was inducted into the Rock n Roll Hall Of Fame. Jeff Ament (the bassist of the band) wore a t-shirt with all Bands/artists that he loved and the Rock n Roll Of Fame was ignoring them. On that list, there was Bon Jovi. (Interestingly, Bon Jovi was inducted the following year.)
The T-shirt:
https://www.instagram.com/p/BSor1Cmg...ource=ig_embed
Do not forget that all artists have an ego and an opinion. Reminder video: Something For The Pain. A video that was basically poking fun at the grunge music and hip hop of those years, including an Eddie impersonator.