To the best of my knowledge, having the health monitor on will not trigger auto runstats, but I admit to not being an expert in either of these "features." It may warn you about how long it has been since runstats were last run, but I don't really know about that either.
Either of these features can cause a lot of problems (including instance crashes depending on which release and fixpack you have), and I prefer to write and run my own scripts to do these things. These features are basically a marketing ploy to convince prospective buyers that DB2 doesn't need much in the way of DBA support; however, there are a lot of kinks that need to be worked out before I would use them.