Bringuier was born in Cuba and studied at the University of Havana where he qualified as a lawyer in 1957. As an opponent of Fidel Castro and his government, he left Cuba on May 4, 1960 and moved to Guatemala; he also lived in Argentina for a brief time before arriving in the United States on February 8, 1961. He set up home in New Orleans, and opened a clothing store called "Casa Roca".
Bringuier joined the Directorio Revolucionario Estudantil, the Student Revolutionary Directorate (DRE), an anti-Castro group. Bringuier was made New Orleans delegate, and placed in charge of DRE publicity and propaganda. Bringuier's clothing store in New Orleans became the local headquarters for anti-Castro Cubans. On August 5, 1963, according to Bringuier, Lee Oswald visited him at his store and posed as a friend of the Cuban exiles, offering to join the fight against Castro. The next day, Oswald again visited the store and left his Guidebook for Marines for the absent Bringuier. Bringuier testified to the Warren Commission that he was uninterested in Oswald's offer.
On August 9, 1963, Oswald was spotted by Celso Hernandez (a friend of Bringuier) handing out pro-Castro Fair Play for Cuba leaflets at the intersection of Canal Street and St. Charles. Hernandez told Bringuier of Oswald's leafleting and the two of them, along with another anti-Castro militant, Miguel Cruz, decided to confront Oswald over his duplicity. As the Cubans accosted Oswald, a crowd began to gather. Bringuier attempted to incite the crowd with his story that Oswald had tried to join his anti-Castro movement and that Oswald was actually a communist and supporter of Castro. Hernandez grabbed Oswald's leaflets and a fight broke out. Oswald and the three Cubans were arrested for disturbing the peace. After the arrest, Bringuier and his Cuban friends were able to post bail, whereas Oswald was not. Oswald was found guilty, fined $10, and released. Before leaving the police station, Oswald asked to speak with an FBI agent. Agent John Quigley arrived and spent over an hour talking to Oswald.
To garner information on the New Orleans chapter of the Fair Play for Cuba Committee, Bringuier said that he sent his colleague Carlos Quiroga (who posed as a Castro sympathizer) to question Oswald. (Oswald had established the New Orleans chapter of the FPCC in May 1963 and was its sole member.)
Bringuier then called William Stuckey, a reporter, who hosted a weekly radio program called "Latin Listening Post" on station WDSU. Bringuier informed Stuckey of Oswald's background and Stuckey went to see Oswald, persuading him to be interviewed on radio; the interview was recorded on August 17.
A few days later, Stuckey invited Oswald to take part in a radio debate with Bringuier and Edward Butler, executive director of the anti-Communist group Information Council of the Americas (INCA). The debate took place on August 21; both Stuckey and Edward Scannell Butler 3rd had in the meantime discovered the story of Oswald's 1959 defection to the Soviet Union and confronted him with it. During the debate, Bringuier challenged Oswald to agree with Fidel Castro's denunciation of President Kennedy as a ruffian and a thief.
Bringuier was pleased with the result of the debate and issued a press release, warning the public about Oswald. After Oswald was arrested for the assassination of President Kennedy on November 22, 1963, Bringuier gave his account of Oswald to the FBI and other authorities. He testified before the Warren Commission and, in 1967, was involved in Jim Garrison's investigation of the assassination.
Bringuier is the author of two books: Red Friday and Operation Judas, the latter available only in Spanish. He was played by Tony Plana in the 1991 film JFK.