I don't see where the cultural issue comes in here, as a Japanese daughter-in-law will still have been brought up to respect her husband's mother and not say the kind of things your DIL has.
So forget her Japanese background.
If she refuses to speak to you, there is little chance of a frank discussion of what in her eyes you did wrong.
I would try to have a conversation with her and your son, preferably without the children around, and say frankly that you are sorry if you offended or hurt your DIL, but you felt that she was trying to run your household for you, which is not something you are prepared to put up with.
On the other hand she does have the right to give her children what she considers healthy food - if she pays for it, but she cannot demand that you have added expenditure to cater for her children ( if this was the issue, which I do not know)
"Negative speak" is not English, so a definition is necessary. If like many modern parents she is opposed to children being told off there is a difficultly, because you like me and most of our generation prefer well-brought up children.
I am sadly afraid you cannot build bridges, but start by saying you are at a loss to know what really has gone wrong, but you are sorry that this very uncomfortable situation has arisen, and you would like to mend it.
At least you and she can probably agree that this has been the worst experience either of you have ever known. It sounds like a nightmare.
Stand fast on your decision that they need to move out, but emphasise that you would like the issues between you and her sorted out before they do leave.