My brother says that the chances of getting a Shiny Rayquaza are programmed in at the beginning of the game (Emerald). So, if I fight it the first time and it is not Shiny, it will never be Shiny. But if it is Shiny the first time I fight it, it will always be Shiny. Is this True?
Your Brother is wrong. There are people who get Shiny Rayquaza after many days of restarting their games. The chances of getting a shiny however are 1 in 8,192, so you're very lucky if you do get it.
I believe now however in the more recent games it is impossible to legally obtain a shiny legendary in-game. I may be wrong on that however.