Should we install beadboard paneling or countertop first?
There are advantages to doing it either way.

If you put the paneling up first, and slid the counter top back so it overlaps the panel a bit, you'll get a very nice looking edge if properly caulked. 

BUT, doing it this way runs the risk of water running down the back splash, and running behind the counter, creating a rot/mould risk on the cabinetry underneath (particularly if the caulk fails).

If the paneling goes over the counter top, you mitigate the water risk, but it's going to be a lot harder to get a good looking edge.