Together with the NASA meatball, worn by both the astronauts, the red-on-white McDonnell logo is among the first patches associated with the early American manned space program.

It is well known from its appearance in the Mercury and Gemini White Rooms, where, as employees of  the prime contractor for the Mercury and Gemini capsules, McDonnell technicians (lead by pad leader Guenter Wendt)  made last preparations for the flight and 'inserted' the astronauts into their seats.
Another view of the smaller patch as Jim Lovell (Gemini-7) chats with McDonnell technicians on December 4, 1965.
(1961 - 1966)
We got the patches from Donnis Willis. He received them from Bill Moore, a technician who worked for McDonnell at the Cape for many years. Although not flown, we consider these patches of great value in our Mercury / Gemini collection, especially since they were so visible during the program.

Some color: Gordon Cooper climbs aboard his 'Faith 7' Mercury capsule on May 15, 1963.
A McDonnell technician assists astronaut Gus Grissom, following a test on July 17, 1961, four days before his Mercury Redstone 4 flight.
The red-on-white McDonnell back patch was seen until Gemini-12 in 1966, after which North American Rockwell, as prime contractor for the Apollo Command Module, took over White Room activities.
A smaller pocket patch can be seen in this picture, showing Alan Shepard entering his Mercury capsule on May 5, 1961.
McDonnell Douglas
On April 28, 1967, McDonnell merged with Douglas Aircraft Corporation and became McDonnell Douglas. The company stayed involved in the space program: Apollo (third stage of Saturn V), Skylab (conversion of Saturn V third stage into the the actual Skylab space station) and the Space Shuttle (Orbital Maneuvering System, Payload Assist Module). In 1984, McDonnell Douglas employee Charles Walker became the first company-sponsored astronaut to fly in space aboard Shuttle Mission 41D.
In other words
"...For its first year, many at NASA had thought McDonnell was a money-grubbing, incompetent contractor that was shipping half-finished, misgbegotten capsules to the Cape, where they had to be rebuilt from scratch. By the time McDonnell was making the Gemini capsules, it had gotten so good at its job that the Gemini's routinely arrived at the Cape in launch-ready condition. McDonnell had in NASA's eyes become a paragon of corporate vrtue..."
Murray & Cox, "Apollo, the race to the Moon"