They do countless benefits together, and have for years. I can't say that I have ever heard Bruce praise Jon, or his music in words. However several times when they have been in the same cities Bruce has invited Jon and Richie to guest with him on a song at a show -1993, 1999, 2008 and most recently Hyde Park. Unfortunately that didn't actually happen due to time constraints but Jon and Richie were there and the press confirmed that they were going to play together. I doubt that Bruce would invite him as a guest to one of his own gigs if he was uncomfortable with him or his music.

I can't say that I know of Bruce actually being invited to sing at a Bon Jovi gig though. Perhaps he was invited and turned it down - who knows?

All these guys have known each other and played together for years. Remember, Little Steven guested with Bon Jovi on the 1995 tour. Bruce's keyboardist Roy Bittan played on the original Runaway recording etc.