Watching a German movie and people there keep saying "Kein Stück". Is it some sort of slang? "No pieces" makes no sense. Thank you.
It is some sort of slang, or at least rather informal speech. Literally kein Stück translates to not a piece (no pieces would be keine Stücke) and means as much as "not at all". As Javatasse mentions in a comment, a good translation is also "not even a tiny bit".
There is also ein Stück weit, which is the opposite and means "a little bit", "to a certain degree".
The phrase "Kein Stück" depends a bit on context.
Literally (like in @jonathan.scholback extended answer) in questions like "Would you want/like..." it can be used as a direct, a little bit rude, answer of "Not at all".
In a negotiation this phrase changes its meaning. For example on the question "Could you agree a little bit more with me that...?" ("Würden Sie mit mir einhergehen,dass...") the phrase becomes more like "Not a chance". Switches from meaning "I do not want anything" to "I will not budge".
I'd like to extend Jonathan's answer of "ein Stück weit" to the comparison layer. It's common to use "ein Stück" and "kein Stück" for comparisons (especially if slight deltas are focused):
"Du bist kein Stück besser als er." --> "You are not at all better then him!"
It's also worth to mention the equivalent version with "My" (semantically quite the epsilon from differential mathematics):
"Du bist kein My besser als er".
Further on, the older and not that common "Deut" can be used for many scenarios too:
"Du bist keinen Deut besser als er" --> "You are not at all better then him!"
But in contrast to "My", its common usage fields are a bit reduced. So it's not common to say "Wir sind keinen Deut weiter gekommen". --> "We didn't proceed any further." Maybe this could be the content of a separate question in doubt here.
It's also common to use it within imperative structures that yield quite harsh or absolute aspects: "Keinen Schritt weiter!" --> "Do not move any further!"
The phrase kein Stück means "not at all".
In this phrase, Stück has the meaning of "a certain distance" or "a little bit". The phrase kein Stück – literally "not even a tiny bit" – is the opposite of ein Stück weit, literally "a small distance", which means "somewhat".
Ein Stück weit vertraue ich ihm. (I trust him a little. / I trust him somewhat.)
Ich vertraue ihm kein Stück. (I don't trust him at all.)
English phrases such as "I trust him only as far as I can throw him" have the same underlying concept of trust measured in distance.