In fact, the opposite of your statement is true: Assuming the axiom of choice, the cardinality of the product X^2 of some infinite set X is always the same as X

In fact, the opposite of your statement is true: Assuming the axiom of choice, the cardinality of the product X^2 of some infinite set X is always the same as X. See https://math.stackexchange.com/a/2464655 https://math.stackexchange.com/a/2464655 for a quick proof