Can I use Avid olive and connector insert in a Shimano brake? Basically, this is the question:
I can use the olives and connector inserts proper for Avid in a Shimano Deore M596 brake/hose?
Has anyone tried and can report if it worked perfectly or if there was any problem?
Thank you very much.
No you cannot.  They are not cross compatible.  Shimano Olives, hose and inserts only work with Shimano components. The same is true for Avid brake equipment.

Beyond not being designed to work together you void any warranty claim by not using compatible equipment.  You also give up any legal approach to the manufacturer if you use other company's parts.  So if you crash because the brakes failed and want to say that's Shimano's fault and it actually was.  They will say it had Avid olives and inserts on it.