To Wyrm's point, officially DocWagon only works with SINners. Therefore if you're a criminal on the run, they are obligated to turn you in. Or at least to turn in you SIN identity. For an example of what I mean, consider this:
DocWagon has a contract for SINner David Evans. David Evans is the pseudonym of a runner named Edge, who is actually SINless. In the eyes of DocWagon, Paul Hewson is hurt and brought in to a DocWagon hospital. While Mr. Evans is unconscious, DocWagon gets a call from Aztechnology saying that Paul Hewson is a wanted fugitive. DocWagon begins to go through the motions with Aztechnology, getting all the paperwork in order for extradition. Meanwhile, Edge realizes he's going to be in deep shit, so he changes his SIN over to Adam Clayton. DocWagon comes into David Evans' room and finds Adam Clayton there instead. Adam Clayton gets kicked out of the DocWagon hospital for not being David Evans. Aztechnology comes to claim David Evans from DocWagon, but DocWagon has no idea where he's gone. David Evans has disappeared.
In game terms, this would mean that the character didn't get fully healed and only healed as much as DocWagon would let him heal before they kicked his ass out of the hospital. I should also point out that this sort of trick should only work if the character regains consciousness, the player thinks of this situation, and the character has multiple top-quality SINs to prove his identity. Assume that the hospital staff has no idea who this guy is, because after treating dozens of patients and changing shifts a couple times, the nurses and doctors have only the paperwork to go on to know the patient in question.
Also, if there's ever a question of what to do with DocWagon, just assume it's like American hospitals. The doctors there are obligated to heal you, but they aren't obligated to keep you out of the hands of the cops. If they even suspect you of a crime, they'll just handcuff you to the bed and continue to heal you. The big difference is that DocWagon doesn't care how you got hurt. In American hospitals, if you show signs of being shot, they have to report it to the authorities. Not so with DocWagon - they assume you're going to be wounded by any of the thousands of things out there that can kill you (read: SINless folks with guns).