There is no problem bringing your own gun to Argentina and most avid hunters will want to do just that. The particular adjustments to your specific gun and the ability to practice before your arrive make this highly desirable. However, it is very common to rent our guns if not for your primary gun then as a back up to your personal gun. We are partial to the Beretta 391, we find the lightness of the 20 gauge will go easier on your shoulder after you have hoisted it a 1000 times in a day. We also use the Benelli M1 Field gun and have great success with it. If you shoot a Browning B-80 it is basically the same gun as the Beretta and should do well. Older Beretta 302’s and 303’s will give you equal satisfaction if they are in great shape. Browning Golds or Remington 1100s may work well but you will want to be sure you have any spare parts along with you.
The bottom line is that you will want to bring a gun that can take at least 5000 rounds without leaving you grumbling and so black and blue you don’t enjoy the trip. We clean the guns daily so they are in top form.