Yes! You totally can!
I think we all got in such a habit of dying white eggs that we've just been convinced that brown eggs won't work. But they definitely do, and I think they make some of the prettiest colors. You have to let them sit in the dye a little longer to get the color to really soak in, but the wait is worth it. You end up with beautifully rich colors that I think look more "earthy" than the same store bought egg dye applied to white eggs.
We did a little comparison - these eggs were all dyed with the standard old store bought egg dye, mixed with vinegar according to the package instructions for vibrant colors. The group on the left are brown eggs (except the blue row), the group on the right are white eggs. (and all the little white marks are just where my kids drew on the eggs with the wax crayon).
Brown eggs are usually more expensive, so I don't know that I would go out and buy them just for Easter egg dying. But if you have them and weren't sure if you could use them - go for it! I absolutely love how they turned out.