Yes, pirates may have worn eyeliner. Sort of.
Although we probably recognize it more from Johnny Depp's portrayal of Captain Jack Sparrow in Pirates of the Caribbean, a form of eyeliner made from lead sulfide called kohl has been used since ancient times.
It's primary purpose was to protect the eyes from eye disease and sun glare. It was also believed to ward of "the evil eye" in some cultures.
Kohl was used by the Ancient Egyptians, and is traditionally worn by women, men and kids in dry or hot places like North Africa, the Middle East and South East Asia, even to this day.
Based on this, it would make sense that pirates, who are exposed to sun for long periods, would take similar precautions.
However, it's unlikely that it would have looked anything as neat as Jack Sparrow's eyeliner, and my guess is pirates applying kohl would have more closely resembled "NFL warpaint".
I thought this comment on Gmefaqs.com sums up the idea of pirates wearing eyeliner pretty well:
"This is correct. And it's not eyeliner -- it's kohl. It's an ancient cosmetic that was mainly used in Arab culture to prevent eye ailments and sun glare, and it was used by both men and women. It makes sense for pirates to wear kohl considering they spend most of their time out at sea, and water attracts the most sunlight. Johnny Depp's research for his role in PotC led him to discovering that pirates wore kohl (along with grease and soot) to shield their eyes from the sun, and this is why we see Jack Sparrow wearing "eyeliner" in the PotC series. And he isn't the only one, Blackbeard in PotC: On Stranger Tides can also be seen wearing kohl around his eyes.
Besides, in an era where Englishmen -- the ones in power no less -- wore powdered wigs and makeup, is it really that hard to believe that some pirates wore "eyeliner" as a sort of primitive sunblock?"