Yes, the probability of drawing the black rock is 1/6 for each person.
This is obvious for the first person. Intuitively "by symmetry," it
should be clear that everyone has the same probability 1/6 of drawing
the black rock.
One more formal way to look at it is that there are $6! = 720$ orders
in which the six rocks can be drawn. Let's number the rocks 1 through 6,
with #1 being the black rock. Some examples among the 720 ways are
654132, and so on.
Suppose you are third in line to draw. Then you would get the black
rock for outcomes such as
How many ways are there to arrange the rocks so that the black rock is
third? Put the black rock in the 3rd position, and then there
are 5! ways to arrange the other five rocks. So the number of ways for
you to get the rock are 5! = 120.
Then the probability you get the black rock is $5!/6! = 120/720 = 1/6.$
Addendum: While I have been typing this and taking holiday phone calls,
I see that two other answers have appeared. I have up-voted them both, as
being reasonable alternative methods to show the same thing.