The Asian flu pandemic of 1957 claimed roughly two million lives worldwide making it one of the three deadliest of the 20th century. It reached the United States in the summer of 1957 but that initial wave caused few casualties. However, the second go-around in March of 1958 was responsible for upwards of 70,000 deaths.
According to this news article from October of 1957 the flu was widespread in Detroit but accounted for only two deaths. Small outbreaks occurred for the next nine years but not to the deadly extents of the early waves. By 1967 the H2N2 strain had evolved and disappeared. Yet, the great vaccination machine kept rolling. And right on cue as the 1968 outbreak of the Hong Kong flu kept the pharmaceutical companies in the black for decades to come.
This unsigned card for one 1.0 cc. dose, subcutaneously, of Influenza Vaccine, Types A and B was issued by C. D. Barrett, Jr., MD, of the Detroit Department of the Health. Since it doesn't state who was the employer it's probably wise not to speculate.
One thing to note is that I found this in a collection of photos and documents at an estate sale in Livonia a few days ago. Among the papers is a Wayne County Emergency Relief ID slip issued to a Peter Ferenc in 1934 which may or may not be related to this card.