From classes that I've taken, you can not just use statistics to imply causality, you have to perform a repeatable experiment that shows that A actually causes B.
Statistics can be used as the basis for performing an experiment though.
So that data could then be the impetus to perform an experiment to see if sleep does cause sickness.
Something like having groups of people randomly assigned to groups, where they are made to sleep very specific lengths of time, and then seeing if anything happens to them health wise.
This can then be used to see if A actually causes B.
As opposed to
B causes A (people who are sickly, sleep longer)
Or there is a third factor, C, which causes both A and B. (People with stressful jobs are more tired, and thus sleep longer, and suffer from more stress induced sickness.)
There is an example of this that was presented in a course I took:
The crime rate increases with increased sales of ice cream.
This would imply that Ice cream is the cause of increased crime. The real cause being that in summer months, people buy more ice cream, and that crime also increases in the summer.