toenail question

Pull it off if you can do it easily, but don't 'tear' it off. 

yea, give 'er a good yank.  You want it off.  I had a similar incident, except mine was broken behind the cuticle from someone stomping on my foot.  I didn't remove it and ended having the new nail severely ingrow on all 3 sides.  After masive infection and 3 exceedingly painfull surgeries (local anestesia, could feel them cutting the flesh away and scraping along the bone), it's finally to the point I can tolerate it.  Still have to dig out little pieces of nail from time to time.

If you keep having issues, find shoes that are a touch longer.  From everything I've heard, the cause is the seam in the toe box catching on the lip of the nail and gently lifting it with every step.  So if you move that seam away, so it doesn't hit, or go in and trim the seam down, it won't catch and no more black nail.  The other cause is your foot smacking the front of your shoe, shoving the nail back into your toe.  You get this alot going down hill.  Again, longer shoes and make sure your laces are tight enough to keep your foot from shifting.   When I was backpacking, I even went so far as to lace (and tie) the middle third of the boot tighter then the rest, so there was more restraining force keeping my foot back, but not a lot of added tension around the rest of my foot.  Anyway....  good luck solving your problem.