So "Black Syphilis" is more or less a urban myth to stop solders from messing with the local girls.

There are cases of possible bacterial infections that mimic "black Syphilis" supposed symptoms, which might be the route of the rumours but generally speaking it's a made up disease.

Here is an interesting article regard STI and the propaganda behind them (including black Syphilis) http://www.psywarrior.com/PSYOPVD.html

Edit: No there is no truth behind Black Syphilis.

Edit 2: since people are asking I'll pull out my nurse hat here. The likely (sorry no definitive research I can find on short notice) bacteria is Staphylococcus aureus, which in rare cases can in rare cases lead to Necrotizing soft tissue infection (necrotizing fasciitis, soft tissue gangrene) which includes skin that changes into a painful brownish, purple and finally black patches.

Edit 3: Also one infection not mentioned but did exist prior to, during the war, and continues to be a on going concern in over populated areas with poor sanitation is Yersinia pestis. Aka the Bubonic plague also known as the "black death" it's characterized by inflammation in the lymph nodes that thanks to the infection often turn black, and break open and leak pus/blood. Which could also be another source behind the "Black Syphilis"

Edit 4: Since several people have stated it would be Fourier gangrene. I'll copy my response from below:

I'm resistant to calling it Fourier gangrene as there are several differences including the age range and causes. Usually Fourier has long Standing blood vessel and nerve damage in the  perineal, perianal, or genital areas that is catalyst after trauma. While some urologists group any necrotizing soft tissue infection as Fourier most only do so if it meets all the criteria.

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