Yes, you can set up conditional formatting on the button. Of course, you would first need a way to determine who the current user is. This can be accomplished, for example, by querying active directory. Do you have a way of doing so? If yes, then query active directory to determine who the current user is, and then set up conditional formatting based on that. For example, If CurrentUser is not equal to jsmith, then hide this control.
Good luck!