If I'm envisioning this correctly, you are applying a 1/4" panel to the back of a plywood frame.

I'd recommend a wire nail rather than a brad. The wire nail has a larger head that will resist pulling through the beadboard panel better. 5/8" to 3/4" long and something like a #16 - #18, depending upon what your hardware store carries. example

If you are able, I'd make this so you can set the panel into a 3/8" x 3/8" rabbet all around the back. This will both look neater and help resist racking.

Thanks! Yes, you were envisioning it correctly...

I love the rabbet idea, but I think it's too late...all the shelves and the benchtop are already in place and I'm just putting the back on...maybe next time!

I was considering stainless steel shanked siding nails, but they all seemed too long...

3d finish nail.