I'm not sure what it means to "check the status" of a server... but if one of the possible values for status is "Not Running" then it doesn't make sense to have something running in that server try to check for that, right?
You should run something outside the server to do the checking.
And no, that wouldn't be a servlet. A plain old
Java application run by a plain old job scheduler would be more like it.
I'm not sure you can get from just knowing whether some servers are running to a proper load-balancing application for those servers -- I certainly wouldn't try writing that myself -- but that's a separate topic. Perhaps you aren't actually as ambitious as what that description sounds like.