Not necessarily true. There are risks associated with new employees, as well as costs for training. Plus, it could mean that deadlines are missed. So basically, by giving your boss an ultimatum, you are forcing your boss to consider whether or not it is worth the costs and risks to get rid of the "squeak." A new cog could be even squeakier than the last.
With that said, pick your freaking battles for goodness sake. It's good to have a job in these times; consider your options carefully before giving it up. Plus, you don't want to jeopardize the relationship between you and your boss by giving an ultimatum. Look for better solutions; that should come naturally to good programmers anyways.