The Green Party nominee for U.S. House who received the highest percentage in the nation (when districts are excluded that did not have both a Democrat and a Republican also in the race) in 2012 was Ursula Rozum. She was the Green Party nominee in New York’s 24th district, centered on Syracuse.
She got a great deal of publicity when she received campaign contributions from a wealthy donor who had also given hundreds of thousands of dollars in independent expenditures to various Republicans. She responded by giving those contributions to charity, instead of using them to advance her own candidacy. She also received publicity because she performed well in a televised debate with her major party opponents. She received 7.74% of the vote, or 22,670 votes. The Democratic nominee won the race in that district. See this story about the final election returns.
Two other Green Party U.S. House candidates, both in Illinois, also did well. Nancy Wade, in the 5th district near Chicago, received 5.68% (15,359 votes) and Paula Bradshaw, in the 12th district in southwest Illinois, received 5.61% (17,045 votes). Because each received more than 5% of the vote, the Green Party is ballot-qualified for 2014 in just those two particular districts, but not the remainder of the state.
Among U.S. House candidates who had only one major party opponent, the highest percentage for any Green was Rebekah Kennedy, who received 16.01% (39,318 votes) in the Arkansas 3rd district, running against a Republican and a Libertarian.