It might be another example of Slavic influence on Romanian. I don't know, I'm just guessing, but Czech for example does this frequently, e.g. hovězí maso literally = beef meat, even though the word for the meat is entirely different to the word for the animal (kráva = cow) and would suffice on its own.
These mean the same and can be used interchangeably. Just differently put together grammarly wise.
I'm not sure if I can explain it... I'll give it a go.
In 'Don't you eat...?' the 'not' is combined with 'do'. The negative way of asking put together with the positive action 'eat'.
In 'Do you not eat...?' the word 'not' is linked to 'eat' making the action negative, 'not eat'. So, the question is asked positively 'Do you...?' about some action that is already negative, 'not eat'.
It's quite complicated, and not really of importance in regards to the meaning.