The Pharisees claimed Jesus healed people by the "beezlebubs" power, and often people sought "sorcerers" to heal their ailments, and still do. I don't think Satan "heals" people though, but he can... fabricate wellness? Everything the devil does is deception, and always has a price that is never worth paying. People claim to feel better after doing certain rituals, but its temporary, and certainly never ever worth doing, for even if your ailment is suppressed till you die, you will get that, and many times worse in hell. Not too mention all the other bad things that can happen in your life. Sin is a disease, and once you feed it, it starts a chain reaction that causes self-destruction.
Short answer is: No, he can fabricate wellness or suppress illness at best. But the preacher probably chose his words poorly.
Rely on Jesus and no one and nothing else. (Obviously)