Pour 1 cup of non-dairy beverage and the water into the rice cooker inner pot. Press White Rice and allow mixture to come to a boil. Stir in salt and rice. Cover and let simmer for 30 minutes.

In a small bowl, whisk together remaining 3/4 cup non-dairy beverage, sugar, egg substitute and vanilla. Stir pineapple with juice into rice cooker. Stir in egg mixture and rum. Let cook, stirring frequently, until mixture simmers and thickens. Pudding will thicken somewhat as it cools. Serve warm or cold.