Yes.
Alyson Hobbes, a Stanford historian, recently published a book called A Chosen Exile:A History of Racial Passing in American Life , in which she recounts a history of "racial passing" in American history, where people who were born ethnically and culturally black, chose to pass as a white person culturally in order to take advantage of the privileges that have historically come with being white in America.
Firstly, you have to remember that the documentation that comes with being a citizen is often a relatively new development, or was inaccessible for the most part. For example, Social Security numbers are not even 100 years old yet, having only been created in the late 1930's (the first Social Security benefit check was mailed in the ancient year of 1940). Birth certificates were not mandated until 1902. For the most part records of birth and death were handled in church registration records and those weren't standardized either, for the longest time. So there was for the longest time a relative inability to be able to document someone's history. There were no computers or ability to quickly transmit records either. So verification of identity, was a bureaucratic nightmare and often only carried out when it was a legal necessity of great importance (wills and estates disputes for example).
Because of this, the documentation required to actually enlist was surprisingly sparse. For example, if we look at President Harry Truman's enlistment paperwork from 1917 is all of four large font pages long. You'll note they're more concerned that the recruit swears he's not lying...and he's sober. Now if you look at the paperwork come WWII, you'll see that they are a bit more stringent in their requirements., but they are still relatively lax compared to today. A certificate of baptism passed as a birth certificate, and if no records were available, you could have someone sign a voucher saying, "They are who they say they are," such as Audie Murphy did when he enlisted and lied about his age, getting his sister to vouch for him. In fact, such cheating of proof of birth was so common during WWII that there is an actual Veterans Group for underage enlistees. If there was a documentation issue, it would be relatively easy if you knew where to get it, a fake document, or five bucks (a decent amount of money considering a cup coffee cost at most a dime) to have someone say, "Yep, this is my kid," to the recruiter.
So, yes, concievably, one could fib their way in, there was no actual way to verify, and by the time WWII came around the U.S. military swelled from half a million in 1940 to almost 15 million in five years, they weren't picky.
Additionally, we have proof of it happening...
While everyone thinks that James Merideth was the first black student at Ole Miss we know of was actually Harry S. Murphy. According to Murphy, when he walked into the enlistment office, the recruiter simply checked "W" for white cause he had light skin and wavy brown hair. Murphy never corrected them. He was now white. From there, Murphy was sent to Ole Miss as part of the V-12 program that sent recruits to universities for an education and officer training. So Murphy, hiding as a white man in the Navy, because a Naval ROTC cadet at Ole Miss.
So yes. It was completely possible for a black person in America during the era of Jim Crow to enlist or be drafted into the U.S. military, but due to "passing features" became white.
At least a few people did. Anatole Broyard, of mixed-race Louisiana creole background, served as a captain in the Army during World War 2 (ironically, commanding an all-black labor battalion). After the war he would continue to pass as white, becoming a writer and literary critic for the New York Times.
Allyson Hobbs' A Chosen Exile: A History of Racial Passing in American Life covers several example's of "passing". One of which was Harry S. Murphy, who was marked down as white while enlisting in the Navy and chose not to correct them. After the war he briefly attended the University of Mississippi as an ROTC cadet before transferring to Morehouse, a historically black college, and living as a black man again. When Ole Miss fought to keep James Meredith from becoming their first black student in 1962, Murphy came forward to the press to let them know that proverbial ship had sailed over a decade ago.
A slight tangent that may be of interest is how much of a sacrifice passing as white could be for the individual and their family. They would often have to cut off contact with their black relatives, sometime claiming they were orphans. The family they left might also claim the passing individual was dead to deflect questions. They had "passed away", so to speak.
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