I switched my settings on my Nikon D7100 to single point focus but now as soon as I turn my camera on, I have a square focus icon in my viewfinder, not just when I press on back button focus. This can't be right.
When using single point focus, you need to line up your subject and focus point selection for focusing, so the information on selected focus point location is useful even before auto focus has not yet been activated. Thus, on a D7100, the focus point is always displayed in single point mode. This is indeed different from dynamic AF area modes, where the camera doesn't know in advance which focus points it would use and therefore AF points are shown only when AF is active (user manual page 74).
Note that there is a separate "●" symbol in lower left corner of the viewfinder to indicate that focus has been achieved.