The episode is season five's "Social Contract" which gives us Nick, a book editor presenting with Phineas Gage-type behaviour (read up on this guy; his story is pretty interesting) in that he cannot inhibit his thoughts and thus uncontrollably "speaks his mind". After a few guesses and tests, they find out that:
What the what?
Last time I checked, the cingulate gyrus was nowhere near being dangerously close to the brainstem; if anything, the cingulate gyrus is a landmark structure that would be relatively easy to locate surgically. In the below image I've blocked out the relative locations of the cingulate gyrus (in pink) and the brain stem (in yellow). I've also circled in red the section of the brain that was highlighted on the screen in the episode, which corresponds roughly with the anterior cingulate.

As you can see, unless the neurosurgeon at Mercy happens to be Fred Flinstone hacking at sheet rock with a hammer and chisel, it would be fairly unlikely that he would come into contact with the brain stem while working on the cingulate gyrus, especially if it's the anterior cingulate (they're at completely different poles of the brain!).
Okay, so I do realize that it can be tricky to always get the medical terminology exactly right, and that all of this mumbo-jumbo is being thrown at the average viewer within the space of about forty or so minutes per episode, but the writers do have medical experts and fact checkers to consult, don't they? Or, you know, encyclopedias? Even a quick visit to the internet would have cleared this one up in a jiffy. For shame.
(References and disclaimers and all that jazz)
House M.D. is owned by Twentieth Century Fox.
Image source: Brain, medial view (Fancy colouration by me.)
Cummings JL. "Frontal-subcortical circuits and human behaviour". 1993. Archives of Neurology. 50(8): 873 - 880.
O'Driscoll K and JP Leach. ""No longer Gage": an iron bar through the head". 1998. British Medical Journal. 317(7174): 1673–1674.