You can use all trains of all participating operators, with very little exceptions. So yes, you can get to Saint-Jean-Pied-de-Port or any other village with a train station.
Burgos is in fact a real city in ESpana with at least some 120k people.
As rian tells, you can board/alight at any station, be it giant big or tiny small, on all 100+ railcompanies that accept the pass.
NOTE that-we warn for that about daily here, both the TGV in FR and even more ALL longer dist. trains anywhere in ESpana MUST be reserved and this may cost a smaller or higehr supplement to pay. SJPdP is only served by local (they call it TER in FRance) trains without, but to get to/from Burgos will always need this.