Rushisaband.com reader Dino pointed me to this reference to Rush in the 2004 Laura Jackson biography of Jon Bon Jovi. On page 9 of the book after a lengthy paragraph citing all of his influences it states:
... But it was after seeing the progressive rock band Rush perform live in NYC that Jon dashed home to Sayreville particularly fired up. ...
Interesting. This statement makes it sound more like he was motivated by Rush rather than influenced by them. I certainly don't hear much Rush influence in his music.