That claim is false, at least in regards to current ball point pens.
Excepts from Pedro Duque's diary in space
I am writing these notes in the Soyuz with a cheap ballpoint pen.As it happens, I've been working in space programmes for seventeen years, eleven of these as an astronaut, and I've always believed, because that is what I've always been told, that normal ballpoint pens don't work in space.
But the other day I was with my Soyuz instructor and I saw he was preparing the books for the flight, and he was attaching a ballpoint pen with a string for us to write once we were in orbit. Seeing my astonishment, he told me the Russians have always used ballpoint pens in space.
So I also took one of our ballpoint pens, courtesy of the European Space Agency (just in case Russian ballpoint pens are special), and here I am, it doesn't stop working and it doesn't 'spit' or anything. Sometimes being too cautious keeps you from trying, and therefore things are built more complex than necessary.