How is a negative interest rate possible? Eurodollars at 100 means that LIBOR is 0%. No bank would pay another bank for the "privilege" of lending money. We can have negative real interest rates but that's another story.
Negative LIBOR has happened before, in Switzerland and Japan. Moreover, with a negative CB depo rates, anything is possible. Finally, don't confuse LIBOR with the actual rates at which the banks fund themselves. In answer to the original question, no, I wouldn't buy 'em.
Options are a different kettle of fish entirely... There's absolutely no reason to believe that par calls can't be worth a few ticks. That has happened before in both Eurodollars and Euroyen (in 2001, if memory serves).