'My mind has gone blank' is the correct phrase. Never heard of the other one before..

My mind goes white is wrong

I would say, in British English at least, that one is correct (blank) and one is incorrect (white). And if I didn't know from the context you give here, I would have no idea what you mean by "my mind goes white" - that doesn't sound like correct English to me (more like a literal translation from another language).

In my experience 'my mind goes blank' is an expression use commonly in English. I have never heard anyone use the second one 'my mind goes white', but it may be a more unquie term that a writer or speaker came up with. I would interpret it as they same as my mind goes blank, though.

I assume we are living in a world which is constantly under threat of nihilism. I will never say my mind goes blank in reality, for that simply means I give up myself to nihilism. The state of my mind is based on how I think, and everything else is just a sign of insanity which is not worth thinking.