Word/short-phrase for "word that should not exist"
Your question seems to ask for an adjective (JEL's spurious not only works well but is fun to use), a verb, and a noun. Here are a verb and two nouns for your fill-in-the-blanks.
He referred to the man by a term so vile, the word itself should be exorcised.
exorcise -- to get rid of something evil, troublesome, menacing, or oppressive -- http://www.merriam-webster.com/dictionary/exorcize
After six hours of listening to corporate double-speak, he came to regard every buzzword as an abomination, invented simply to annoy him.
abomination -- something that causes disgust or hatred -- http://www.merriam-webster.com/dictionary/abomination (... and, often, thought of as a defilement of nature)
After six hours of listening to corporate double-speak, he came to regard every buzzword as anathema, invented simply to annoy him.
anathema -- someone or something intensely disliked or loathed -- http://www.merriam-webster.com/dictionary/anathema (... and, sometimes, just plain evil)
Perhaps mistakenly, I've based this answer on the assumption that you are asking for a term that is the converse of sniglet. Also, I assume that by should not exist [in dictionaries], you intend that the word should not be in dictionaries because of objective, evidentiary reasons, rather than because of subjective, emotional reasons. So, I'll provide
a word/short-phrase for a word that does exist [in dictionaries], but shouldn't [because it's not a word].
The term for "a word that does exist [in the dictionary] but shouldn't" is 'spurious', or 'spurious word'. I was fascinated to discover a list of these apparent contradictions in my free 1970s book club OED-with-magnifier.
The terms, 'spurious' and 'spurious word', satisfy your examples:
He referred to the man by a term so vile, the word itself should be spurious.
After six hours of listening to corporate double-speak, he came to regard every buzzword as a spurious word, invented simply to annoy him.
The first 'spurious word' currently provided in OED Online is
ballow, n.
This explanation is provided in place of a definition:
Prob. a misprint for baton n., to which the word is emended in many modern editions of King Lear. The Quartos read battero (which is prob. another spurious word, perh. arising from a wrong expansion of a typographical abbreviation of batton, variant of baton n.) and bat bat n.2 The word is given in later dicts. (e.g. Bailey (ed. 3, 1726) and Grose (1787)), based on the occurrence in the quot., but does not appear to be attested independently. For an alternative etymology, see E. Ekwall in Eng. Studies (1941) 23 99–101.
["ballow, n.1". OED Online. December 2015. Oxford University Press. http://www.oed.com/view/Entry/15006?rskey=GkvnKr&result=1&isAdvanced=true (accessed January 09, 2016).]
About thirty other 'spurious' or 'probably spurious' words are provided by OED Online. They include such delicacies as
disgore, v.
Spurious word in Ash, etc.: see disgorge v. 3.
(OED Online)
evacate, v.
a spurious word in Dicts; see evocate v.
(op. cit.)
enhendee, adj.
A spurious word found in some heraldic and other Dicts. in the phrase cross enhendee (given as synonym of cross potenee) where the adj. appears to be a corruption of Old French enheudée having a handle. The misreading occurs in French writers, e.g. Palliot, 1664.
(op. cit.)
fructiculose, adj.
Spurious word in mod. Dictionaries: see fruticulose adj.
(op. cit.)
Your two examples have very different connotations. So words like taboo or execrable or pernicious work well with the first, but jargon or drivel or mumbo-jumbo work better for the latter. It is hard to slot in one phrase. You also complicate things by requiring a noun for a single word, not a noun for a manner of speaking. (You don't say "a balderdash" or "a bullshit," but you might be able to say "an anathema." That's been answered already, though.)
Taboo works well as a noun or an adjective:
taboo - banned on grounds of morality or taste <the subject is taboo>
So you could say you regard a word as a taboo.