The full flight profile of a VTVL Grasshopper-like vehicle hasn't been shared by SpaceX, but it certainly won't "stay vertical" for the entire flight — assuming that vertical is defined as 90° flight path angle.
Ground launch rockets typically start by flying more or less straight up. Because they are at low angle of attack this keeps lateral aerodynamic forces manageable and it gets the rocket out of the thick lower atmosphere as quickly as possible. But as soon as possible (exactly where depends on all sorts of parameters but above 100,000 ft would be an order of magnitude altitude) they begin what's called a gravity turn. The trajectory begins to turn over and the flight path angle decreases.
So it is likely that the vehicle will have started to turn over by staging and thus will no longer be vertical.
Assuming the vehicle falls ballistically before restarting the main engines for landing, there are really only two high-level options:
- Actively control attitude: use an attitude control system to control how the vehicle is pointing
- Let the vehicle attitude float (i.e. not controlled).
Which is best to choose would be a trade buried in the details of the rocket and trajectory design, and which we'll only know the answer to when SpaceX decides to share it.
Either way, the vehicle will have to be properly re-oriented for engine ignition before it got too deep into the atmosphere. Depending on the aerodynamic stability of the vehicle this could be done in part passively or with (or augment with) active control. Because it's only a first stage it will only be going several Mach, and thus probably won't need a thermal protection system as heavy, complex, or expensive as what was used for Shuttle or even the Dragon or Apollo re-entry vehicles.