Peterson is notorious for using imprecise language and failing to provide sources for claims. For example, there aren't multiple "psychophysiological systems" - there's just ... psychophysiology. Similarly, there aren't well-defined psychophysiological response patterns of "challenge" or "withdrawal" (though see fight-or-flight for something vaguely related). However, in lectures like this, Peterson is playing the role of a motivational speaker, rather than a science communicator, so there is no expectation that evidence should support his claims - basically, if his presentation motivates you in a positive direction, then enjoy it. Anyway, let's just ignore Peterson's description then, and ask more broadly:
What happens when people undertake stressful challenges either voluntarily or involuntarily?
Well, as is often the case with broad questions like this, the answer is ... it depends. This question is common in the sphere of education, where educators contemplate whether to make programs compulsory to increase participation, or whether doing so will reduce the benefits of those programs.
A well-studied example is service-learning, in which students either volunteer, or are required to volunteer, for community service that ideally supports their learning goals. Service-learning can certainly be challenging and stressful, but research suggests that it has a wide variety of benefits for students and their community. How are those benefits affected by the voluntary nature of participation?
Schmidt, Shumow, & Kackar (2007) for example, found no difference:
Participation in any service is associated with positive outcomes
whether service is voluntary or required.
Similarly, Flanagan et al (2015) report:
... we found no differences ... between those youth who were mandated
to do community service versus those who engaged in service
From Henderson, Brown, & Pancer (2012):
Mandated community service experiences are not significantly different
from those entered into willingly.
And similar results from additional studies (eg, Hart et al, 2007; Henderson et al, 2007; Metz & Youniss, 2005). However, other studies did find some notable differences in outcomes (Kackar-Cam & Schmidt, 2014; Horn, 2012; Kim & Morgül, 2017).
There are also studies on voluntary vs involuntary participation in extra-curricular activities that are too numerous to review here, but suffice to say that results do not clearly favour voluntary participation.
- Studies of this nature inevitably suffer from the self-selection bias: Of groups offered the choice, those who have greater competence are more likely to volunteer. Thus, in cases where voluntary participants outperform involuntary participants on some measure, it is not always clear if the reason for this is due to the voluntary nature, or to pre-existing competence. As an example, a study by Rodriguez-Aflecht et al (2017) on voluntary vs involuntary participation in a math tutoring game, found that the volunteer players had higher pre-existing math skills and interest than those who did not volunteer. Nonetheless, involuntary participants outperformed them on many measures.
- Studies of this nature often situate results within self-determination theory (SDT) - a general theory of motivation. SDT predicts that higher levels of autonomy result in greater engagement and improved performance, supporting the assertion that voluntary participation is superior to involuntary. So why does this theoretical prediction not play out in the results exampled above? Well, volunteering is but one small decision in a long series of decisions that participants might make during the course of the program. Indeed, many studies on service-learning find that the program's effectivness depends much more on its content - particularly the level of autonomy afforded to participants within the program, the level of competence that students attain, and the level of social support that they receive - than it does on the initial participation decision.
- Studies of this nature rarely incorporate any physiological measures. For obvious and practical reasons, blood samples are not collected from students participating in service-learning, nor is their brain scanned, etc. In general, psychophysiological studies are typically done in-lab, and use relatively simple, well-controlled manipulations that would not usually be expected to translate well to complex, dynamic, ecological situations. As such, it would be out of context to use such research to support assertions about handling real-life challenges.