beadboard ceiling over popcorn ceiling?
I am thinking you could get by without furring strips, with this in mind, you want the nails to go into the ceiling joists at least 1" or more. So, depending on what you are nailing up, in thickness, and whether you are driving the nail straight in or angled, as in nailing into the tongue of carsiding, you have to decide if you can obtain this 1" of "grab" into the joists, including going through the drywall. 3/8" beadboard has small tongue and grooves whereas 3/4" carsiding has 1/2" tongue and grooves so you can use 15 or 16 gauge nails, 2 1/2" long and fire them at an angle and still grab into the joists more than 1".
Furring strips are more work. I would glue and screw them in place and, as long as you are at it, you should try to level out any wavyness in the ceiling. Once you have done all that, Yes, it will be the superior route to go. That is my take on the 2 choices.