Impossible or improbable? Hurricane crossing the equator
Improbable.

It is well known that the Coriolis force is needed to form a hurricane, and the figure of 5oN/S as the minimum for formation is widely publicized. You can also find record of tropical storm formation near India as far south as 1.4oN.  

The problem of crossing the Equator isn't one of hurricane formation though, it is one of hurricane motion.  Due to Coriolis, a hurricane initially moving parallel to the Equator will start gaining a poleward component to its motion, thus moving it away from the Equator.  But, because this is due to Coriolis, if you could get a storm close enough to the Equator, this effect would not be as strong.  This would be an improbable track, but I'm not willing to call it impossible.  We haven't had satellites all that long, and all we can really say is that it hasn't happened since we've been watching.  

If a storm did cross the equator though, what would it do?  Nothing at first, but as it moved further into the opposite hemisphere, Coriolis would be working against the storm and it would spin down, become disorganized and cease to be a hurricane, probably becoming a remnant low.  

A tropical disturbance has crossed the equator.  One such disturbance occurred June 27, 2008 in the Atlantic basin (south to north) that retained its clockwise motion for some time: