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We got one of those bathroom drug boxes. Its the kind that goes on
the top of the wall. When I screwed it on the wall there were no 2x4s
behind the wall so its just screwed to the thin paneling. There are 4
screws. Is this strong enough? The box is 15wide x 24high and
sticks out 4 1/2 inches.

At 15" wide, you will have hard time finding a stud. There is usually 14
1/2" between them and mot sinks/medicine cabinets are centered over the
opening so a recessed cabinet can be installed, as well as running the sink
plumbing.

To answer your question, it all depends. It may or may not hold up
depending on what is in it, how hard you open or close the door, etc. I'd
try to put one or two anchors into the drywall for extra security. There
are many types that will work from the toggle bold to the tap in types that
expand when you screw into them.