"Just fine" means he feels absolutely OK, he has no complaints.
However, the meaning can change depending upon the context and the tone of voice that's used. For example, you go in to work on Monday morning and find that the boss had come in on Saturday and piled a bunch of papers on your desk, along with a note that says this project must be completed by noon. You had a headache when you woke up, and now it's worse. Your boss walks in and greets you, asking "How are you today?" (He knows full well that he's given you a ton of work to do this morning, and he dares to ask how you feel!) So you roll your eyes and say in an exaggerated and sarcastic tone, "Oh, I'm fine, just fine."