Can my son carry sickle cell if neither of us are carriers?

No.: Not possible, unless your tests or your son's test are in error. Sickle cell trait is inherited as an autosomal dominant gene with 50% acquisition. Two have sickle cell disease, a gene from both biologic parents is necessary!

De novo sickle cell: I did a quick search of the medical literature and did not find a case report of someone developing sickle cell trait from a new mutation not found in their parents (de novo). In other words, if both parents do not carry sickle cell trait, the child will not have sickle cell trait. That being said, it is not biologically impossible, just very unlikely.