That's totally nonsense!

You only need a valid visa to enter the US. You don't have to have a valid visa to stay here, but you do need to maintain your legal status. I came to the US with a F-1 visa, and I was in F-1 status for the next 6+ years without a valid visa (because I didn't go overseas). I changed to H1B status later on, still without a valid visa. My F-1 visa was the only visa I ever got.