The shop however described the request as "inappropriate" and refused to make the cake for young Adolf, who has just turned three.
The Campbells, who also have two daughters, JoyceLynn Aryan Nation Campbell, and Honsylynn Hinler Jeannie Campbell (named in reference to the SS Heinrich Himmler), were offered a cake with enough room for them to write their own inscription but they refused.
ShopRite has also refused to make a cake for Adolf's sister Aryan Nation, who turns two in February.
Mrs Campbell, from Holland Township, New Jersey, said: "ShopRite can't even make a cake for a three-year-old. That's sad."
Regarding their controversial names, Mrs Campbell added: "I just figured that they're just names. They're just kids. They're not going to hurt anybody."
Mr Campbell told a New Jersey newspaper: "They say, 'He (Hitler) killed all those people.' I say, 'You're living in the wrong decade. That Hitler's gone.
"They're just names, you know. Yeah they (the Nazis) were bad people back then. But my kids are little. They're not going to grow up like that."
But others, such as Barry Morrison, director of the Anti-Defamation League, said: "Might as well put a sign around their neck that says bigot, racist, hatemonger. What's the differenceVia The Telegraph