FLEMINGTON, N.J. — Ever since Marie Corfield’s confrontation with Gov. Chris Christie this fall over the state’s education cuts became a YouTube classic, she has received a stream of vituperative e-mails and Facebook postings. “People I don’t even know are calling me horrible names,” said Ms. Corfield, an art teacher who had pleaded the case of struggling teachers. “The mantra is that the problem is the unions, the unions, the unions.” Across the nation, a rising irritation with public employee unions is palpable, as a wounded economy has blown gaping holes in state, city and town budgets, and revealed that...