This is actually a very common injury
and, although painful and unsightly, it does usually heal up just fine over time. If you nail has turned black, then I am afraid your coworker is likely right that your toenail may fall off. Alternatively, it may just grow out slowly with the black part gradually disappearing. Whether or not it falls off depends entirely on how severe the damage to the toenail was.
Fortunately, even if the toe nail falls off completely, this is not usually a major problem. Generally speaking, the toe nail will regrow itself slowly, although this may take weeks to months to reform entirely. Also, sometimes, the toe nail may not grow back perfectly smooth like it was before you injured it.
There is no particular care that you need to give to the injured nail. Do not try to peel it off prematurely - let it fall off on its own. If the skin under the nail when it falls off is tender or sensitive, you can apply a small bandage to protect the area, but this is not necessary if it is not uncomfortable.
I recommend talking to your primary care doctor
for more information.