There was a western movie in the early to mid sixties, about the same
time period as "One Eyed Jacks". There was a famous quote in this
film: "Let's bust out of this puke hole!" What was the name of this
movie, who said this line, and what were the circumstances?

The only movie I was able to find with that quote was an X-Men TV
show.  A sound clip of the quote is below.  I found this at an X-Men
website.

Website: http://www.smartlink.net/~falcon/comics/wolvsound.html (clip
#29)
Sound Clip: http://wolverine.x-knights.com/sounds/wolv29.wav

I found this site by searching google for "Let's bust out of this puke
hole" (no quotes in the actual search).
://www.google.com/search?q=Let%27s+bust+out+of+this+puke+hole

I understand if this answer is not acceptable to you, please just
decline it.  But that was the only instance of that quote I could
find.  I searched Google for the quote and a several movie quote
websites.  Thanks for using Google Answers!

xemion-ga

This researcher found the quote, however, it was not from the sixties
western that I am searching for. However, three stars for a good
effort!

George Bush to the Japanese Prime Minister, 1991

If that's the only thing you can remember about the movie, then it's
not exactly a memorable film. Anything else you can recall - e.g.
actors?

This website might help help you.

http://us.imdb.com/Sections/Quotes/