Greer was born on a farm in Stewartstown, County Tyrone, Ireland, and emigrated to the United States in 1929. After working for over a decade as a chauffeur and servant to several wealthy families in the Boston and New York areas, including the Lodge family and several years with Franklin Q. Brown of Dobbs Ferry, NY (the 1940 census has him as a live-in servant of Franklin Brown of Dobbs Ferry and in Greer's Warren Commission testimony he said that he worked for a "private family" in Dobbs Ferry for "13 years" before enlisting into the Navy in 1942), Greer enlisted in the U.S. Navy in World War II, was assigned to the presidential yacht in May, 1944, was discharged on September 18, 1945 and then joined the United States Secret Service on October 1, 1945.
Greer took a role close to Kennedy, and can be seen in several pictures with the Kennedy family. He chauffeured the president on many occasions, including the day of the assassination. Like all agents involved, he has been the target of much speculation and criticism for his actions on that day. Greer, along with Secret Service agents Roy Kellerman, Clint Hill, and Rufus Youngblood, provided testimony to the Warren Commission in Washington, D.C. on March 9, 1964.
Greer retired on disability from the Secret Service in 1966 due to a stomach ulcer that grew worse following the Kennedy assassination. In 1973 he relocated to Waynesville, North Carolina, where he would die of cancer.