The phrase “wouldn't blink twice” sounds to me like a conflation of a “wouldn't blink” idiom with a “wouldn't think twice” idiom.
“Wouldn't blink”, in forms like “wouldn't blink an eye” and “wouldn't bat an eye/eyelash/eyelid” connote not reacting or responding, ie, failing to show surprise.
“Wouldn't think twice” means not having to give something a second thought; ie, not having to reconsider something, because it is so obvious or clear in the first place.
Edit: Following on from Sven Yargs' comment, some Google ngrams results may be of interest. First, a few caveats:
• Google ngrams records the 't parts of contractions wouldn't and didn't as separate elements. A search for ' t is satisfied by wouldn't, couldn't, shouldn't, didn't, don't, etc.
• Near the end of Google's ngrams info page, it says “we only consider ngrams that occur in at least 40 books”.
• Books linked below the graph mentioned below include a few dozen results for ' t blink twice even though a page label says there was only one result.
With those caveats understood, the general trend according to the Google ngrams graph for ' t blink,' t think twice,' t blink twice, is that overall frequency of ' t blink is about the same as that of ' t think twice; both forms have been occurring since the early 1800's; each occurs several thousand times as frequently as ' t blink twice. That is, ' t blink twice negative contractions are significantly uncommon (a few dozen occurrences, most within the past 25 years) compared to ' t blink or ' t think twice negative contractions.