At the time of the Six Days War, IDF/AF fighters used either the Matra 530 or the Shafrir I. Both were crap, and neither is credited with anymore than a handful of kills during their entire sevice with the IDF/AF. It wasn’t until the introduction of AIM-9s and Shafrir II during subsequent years that any significant amount of missile kills were achieved.
The Arabs were using K-13s at the time.

Why did Mirages do so well? For a number of reasons. Better training and piloting skills is one. Another was that although the Mirage was inferior in many respects to the major Arab fighter at the time, the MiG-21, it had its advantanges, which IDF/AF pilots put to good use. This was in large part thanks to the defection of an Iraqi pilot in a MiG-21 during the previous year. Israeli pilots already knew a great deal about the MiG when it they encoutered it during the war.

The Liberty was indeed hit by Mirages. There was an interview with one of the pilots recently, in the Jerusalem Post, here.

What engine upgrade? Finding pictures of Israeli Mirages and Neshers is easy enough, but I guess you’re looking for something specific.