In the novel Three Men in a Boat, Jerome was medically ill. Do you agree?

No, the character-narrator of the book, J., is not medically ill. He may be a hypochondriac, though, and he may just believe that he is sick. Or at least, J. has a quick inclination to become sick when this possibility is suggested to him.
In the first chapter, J. shares the story of his trip to the British Museum as a way of illustrating his approach to life. Here he consults the medical reference book and quickly believes that he is afflicted with almost all of the diseases described in it. Is this even possible in real life? Of course not. As usual, J. overreacts, or he misinterprets information and experiences that come his way. This early story quickly unveils to us some of the personality quirks of this character. He also tells it in a humorous way, setting the stage for many more odd encounters and humorous situations to come. Three Men in a Boat is full of such scenes. It makes for enjoyable reading.