The chance of a shiny being a square shiny that you found in the wild is 15/16. The chance of a square shiny appearing from a shiny you hatched from an egg is 1/16. And vice versa with star shinies. However, if a static encounter is shiny it has the same odds as if it was in an egg, so it would be 1/16 for a square shiny.
Therefore, the odds of a Shiny Regi being a Square Shiny is 1/16, because the Regis are static encounters. The odds of a random Regi being a Square Shiny is 1/65536.