In antiquity, there was no distinction between illness and sin. When Jesus healed the sick, he was literally and tangibly forgiving sins. This is why the Paralyzed man is healed immediately after Jesus announces his sins are forgiven in Mark 2:3-11 (NIV):
Since they could not get him to Jesus because of the crowd, they made an opening in the roof above Jesus by digging through it and then lowered the mat the man was lying on. When Jesus saw their faith, he said to the paralyzed man, “Son, your sins are forgiven.” Now some teachers of the law were sitting there, thinking to themselves, “Why does this fellow talk like that? He’s blaspheming! Who can forgive sins but God alone?” Immediately Jesus knew in his spirit that this was what they were thinking in their hearts, and he said to them, “Why are you thinking these things? Which is easier: to say to this paralyzed man, ‘Your sins are forgiven,’ or to say, ‘Get up, take your mat and walk’? But I want you to know that the Son of Man has authority on earth to forgive sins.” So he said to the man, “I tell you, get up, take your mat and go home. He got up, took his mat and walked out in full view of them all. This amazed everyone and they praised God, saying, “We have never seen anything like this!”
So in other words, yes, he may have been asking about salvation and healing in the same question. We do not have any reason however to believe that the jailer was infirm, so while this is in the realm of possibility it is unlikely he was asking for physical healing. Were he doing so, it is likely his illness would have been mentioned to clarify.