I think we need to look big picture here. First, let's remember the technical definition of cognate. Here's a decent one straight from Google:
(of a word) having the same linguistic derivation as another; from the same original word or root (e.g., English is, German ist, Latin est, from Indo-European esti).
Obvious candidates for 'non-cognates', then, are English neologisms: words coined by English speakers that have not been introduced by borrowing and that have not been borrowed to other languages. Unfortunately, many new words will themselves be derived from the roots or parts of pre-existing words.
For example, one might think of the English idiom "the real McCoy". But "Mc-" is actually 'a Gaelic ancestral name to mean "son of"' and reappears in tons of Scottish names.
So 'McCoy' is not a 'non-cognate' in your sense.
So also with the word 'maverick'. Though it appears to be a better example, 'the surname Maverick is of Welsh origin, from Welsh mawr-rwyce, meaning "valiant hero"', according to Wiktionary.
The poem "jabberwocky" and some of the words in it might be non-cognates, but even here, many of the words are intentional combinations of preexisting words (e.g., chortle, which had no existence before, = snort + chuckle). See http://en.wikipedia.org/wiki/Jabberwocky
Maybe the best opportunity to find non-cognates would be to look at onomatopeic words, like 'meow'. Since these are imitative there is no reason for them to be borrowed from or shared to other languages.
Then there is another set of corner cases where the word looks to be invented in English from non-roots of other words. Slang words would often fit into this category. I am thinking also of words like 'jazz' and 'doo-wop', or Homer Simpson's 'D'oh!'. Unfortunately, these may be cognates, for the reason that they have been borrowed from English into other languages.