I mean, whether or not this stands up legally isn't worth discussing but I don't think this is typically how an ultimatum goes. Rather it's more along the lines of "this or I'll quit", "okay, we're not doing this", "right, I quit". It's not called calling your bluff for nothing.
I agree that her email to the Brain group makes her position at google almost untenable, but that's a separate question. However, it might make her choose not to fight them on the resignation.
