They keep them shut so they won't be driven mad by the knowledge of when people are going to die.
This leads to an interesting conclusion. Brock has always chased women all his life, in what may seem to be sheer desperation. With the Shinigami Eye theory, it becomes apparent that Brock is cursed, much like his past life of Miroku. Both have not chased women out of perverseness, but necessity. (At least of the thought that they need to continue their lineage).
Alternatively, these same people all possess Geasses, specifically the mind-reading type that Mao had. They keep their eyes shut for fear of their Geasses getting out of control and forcing them to hear everyone's thoughts until they go utterly batty.
As an alternative to the alternatively-alternative: they all are infected with the Ganpuku mushi. The stress of keeping their eyes open for more than a few minutes, if at all, caused severe dizziness and headaches. The brief glimpses into the future they see even when their eyes are closed is what allows them to get around without smacking into everything. Of course, it also means that it's only a matter of time before their eyes transform completely, force themselves out of their eye sockets, and infect someone else...
Alternatively-alternatively-alternatively to the alternative, they all have Kaleidoscope Eyes, but want to hide them beacuse they don't want to be labled a Mary Sue.
And just to throw in one more alternative, they all have sharingan, byakugan, or whatever it was that Pain had.