Regice can only be obtained once from the Advanced Path. It can then be obtainable randomly from the Endless Path.
Hatching a Regice egg will not count towards your Shiny Chain. You can obtain a Shiny Regice Egg by completing 1000 levels in the Endless path, though you have an equal chance of obtaining a Shiny Regirock or Registeel Egg as well.
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Regice does NOT evolve.