So are you saying that if a first-round pick doesn't sign with the team that drafted him and becomes a free agent, the drafting team gets a second-round pick? Does it come from the team that eventually does sign him, or is it at the beginning or end of the second round?

Correct. It comes from the NHL, not the team that signs him. I believe the Sabres would get pick 63 because Johnson was the 31st overall pick in his draft year.