There is no way to validly misinterpret "has a negative mass squared" and then you proceed to misinterpret it?
"The square of a negative mass" is a positive number. The square of an imaginary mass is negative, which is what they are talking about. It is clearly possible to misinterpret this. Before someone cries out that an imaginary mass is nonsense, I should point out that imaginary numbers appear in several legitimate places in science. I don't believe the results of this paper are correct, but I'm not about to dismiss them out of hand.