I can imagine many easy hypotheses, but I hope this is an area that there's been at least some considerable academic thought and research on. I'm more looking for those kind of discussions.
Obviously inspired by today's.... historical incident with a highly bowdlerized name.
Student radicalism played a hugely significant role in the history of 19th century Russia. Starting in the 1830s, and really taking off in the 1860s, Russia's growing university system became a hive of radical politics. Russia's larger universities (in St. Petersburg, Moscow, Kazan, Kharkov, etc.) were routinely affected by strikes and student protests - and, less visibly, students played a large role in the conspiratorial groups of this period. The autocracy expressed considerable anxiety over this problematic situation (the pedagogical pathways intended to craft useful subjects were instead churning out political dissidents), as we can see in the government's habit of shutting down universities, firing professors and expelling students in during moments of unrest both at home and abroad. Indeed, in the 1860s, the word 'student' had become synonymous with 'nihilist', 'materialist', and 'revolutionary' in the popular imagination.
Much has been written about student dissidence in the Russian 19th century - its role in radicalizing the general population, importance in the events leading up to 1917, and prominence as a life stage in the bildungsroman of Russian revolutionaries. However, the question of why the universities functioned thusly is a bit trickier, and not widely agreed upon.
I've seen everything from educational/psychological explanations (students possess a proclivity for iconoclasm & grand, salvational thinking) to more microhistorical analyses (such-and-such a professor of especially radical views taught at such-and-such a university and politicized such-and-such a student). However, I think the most convincing explanations look at the issue from a social, cultural, and intellectual ground.
Social/Cultural: In the Russian 19th century, the university was one of the few spaces of semi-public intellectual exchange. Furthermore, after the educational reforms of the 1850s & 60s and the universities were opened up to students of various ranks (raznochintsy) and the sons of priests (popovichi), this was one of the only places where cross-class intellectual and political dialogues could occur. The free-thinking student circles that developed in universities across Russia became primary sites of radical socialization, and can be seen (as they often are in memoirs) as both symmetrical to and constantly interpenetrating with the illicit / semi-licit intelligentsia circles of the empire's major cities. This bleeds into the cultural as well, for the values that defined the political cultures of these small university circles - brotherhood, self-sacrifice, critical thought - would also be present (in varying degrees) in all of the conspiratorial movements of the second half of the 19th century.
Intellectual: Of course, the texts that these students were exposed to in the universities is also of key significance. If you look at radical autobiographies from this period, the university is by far the most common place where the author was first exposed to a certain 'revolutionary canon' that politicized him/her and set them down the path of radicalism. Of course, much of this reading was not part of the notoriously conservative university coursework of the period, but rather became available to the memoirist through the tightly-knit radical social groups of the university setting (see above). Also, the texts of primary importance changed historically - the Schiller, Hegel and Feuerbach of the 1840s would have been replaced by Comte and Büchner by the 1860s, and a decade later students were reading illegal copies of Chernyshevsky, Dobroliubov & Pisarev. However, the university as a space for the procurement, reading, and passionate discussion of banned literature was a constant throughout 19th century Russian history. As an aside to this, I've also been quite struck recently by the constant use of pedagogical language in memoirs & radical texts from this period. Narratives of revolutionary life are painted in terms of educational/personal development, the most dramatic moments of these memoirs are these almost sublime shifts in understanding brought about through reading a certain text or being introduced to a certain idea, and the goal of radical political activity itself (helping the peasantry, working poor, etc. etc.) is itself constantly understood as an essentially pedagogical activity (indeed, a huge amount of these radical students became educators, carrying everything from radical factory lectures to literacy classes in peasant villages. The second-most popular professions were medical in nature - but the 'medicalization' of politics & the physiological language of radicalism during this time is another topic altogether...). I see this going back to both the Enlightenment (man as a rational being, political change as a critico-rational project) and Hegelian metaphysics (the notion of both an individual and society's rational progression to consciousness was hugely popular in the mid-19th century, and has a marked pedagogical thrust)(However, Idealist philosophy could, of course, also be wielded in the service of autocracy or mildly political liberal thought - see the German Burschenschaft and later Right Hegelianism). It would thus be interesting to examine student movements during the French Revolution - a topic I, unfortunately, know nothing about.
So yeah, I'm sorry if my thoughts are a bit disorganized, but I hope it contributes to a larger discussion. For anyone interested in the history of 19th century Russian student radicalism, two classic works (and my main sources here) are:
Gentle reminder to any potential commenters that the 20-year rule is still in place!
So in my area of study (Middle East) the Muslim Brotherhood had extensive student activism organization in the 1940's. Students also participated in the 1919 revolution in Egypt but considering how broad the support was for that across Egyptian society I'm not sure it's quite the kind of counter-cultural dissidence you mean.
Prior to that, the Young Turk Revolution of 1908 involved student societies but the leadership were all older. To be honest I don't know enough about student involvement in the Committee of Union and Progress/Young Turk Movement to comment, however.
Think back to the first university, Plato's Academy. It sat uncomfortably next to the political establishment for years. In a dialogue named after one of the students, Theaetetus, Socrates makes the claim that intellectuals are not interested in cooking food or making beds, they are interested in bigger questions; what is beauty? How do we best live our lives? What is justice? What is good? And at the end of the dialogue he is taken away to be executed precisely because he asked these sorts of questions. He undermined the civic and religious stability of the democracy by being a pain in the arse.