The context is this video at timeline 43:26 seconds .
That's too fancy for me. I don't need no stinking counters.
What does this mean? Is it an American or British expression?
"I don't need no stinking counters" just means "I don't need any counters" combined with a humorous reference to the catchphrase "We don't need no stinking badges." The reference probably carries no particular meaning: it's an instance of the practice of quoting catchphrases for amusement. (For more about this practice, see the paper Harris et al. (2008), "Social movie quoting: What, why, and how?" Ciencias psicologicas, 2(1):35-45.)
Wikipedia has a good history of the catchphrase. It originates with the novel The Treasure of the Sierra Madre (1927) by B. Traven (a pen name; the writer's real identity is apparently not known with certainty). Here's the passage:
Alfonso Bedoya's performance has been widely referenced and parodied, the most well known reference being the one in Mel Brooks' Blazing Saddles (on YouTube here). See Wikipedia for many other references in popular culture. The phrase is usually given not as an exact quotation from the book or movie, but in the compressed form, "We don't need no stinking badges."