It depends on how you order it. With the natural numbers in ascending order, squares are less common. Interleaving them like {1, 2, 4, 3, 9, 5, 16, 6, 25, 7, …}, they’re equally common. With a different order type like {2, 3, 5, 6, 7, …, 1, 4, 9, 16, 25, …}, I have no idea. This is a problem.
It depends on how you order it. With the natural numbers in ascending order, squares are less common. Interleaving them like {1, 2, 4, 3, 9, 5, 16, 6, 25, 7, …}, they’re equally common. With a different order type like {2, 3, 5, 6, 7, …, 1, 4, 9, 16, 25, …}, I have no idea. This is a problem.
See also Nick Bostrom’s Infinite Ethics [PDF].