Yes or No?
A good economics student should disagree with the above argument as it stands. It is not clear that college and university education provides significant external benefits---that is, income to the community in excess of that which is paid to the person holding the education. Even if such external benefits are in fact present, it is unlikely that they would be so great as to warrant free provision of college and university education to all that qualify for entrance into the programs.
What such a policy does is educate the brighter young people in the community at the expense of those who do not choose to so educate themselves or, more important, do not possess the ability to qualify for college and university entrance. Of course, proponents of the policy will argue that college and university education is so costly that only the rich would be able to afford it without subsidization, so the policy is justified on income-distributional grounds. The way to deal with this problem is for the government to loan to students the correct full amount of their educational expenses at an appropriate interest rate and have those individuals pay back these loans in the form of specified additions to their future income taxes. This would require that they not be allowed to leave the country until their educational expenses are repaid, unless they go to a country that has and agreement with their home country on income-tax deductions. This policy involves, of course, a subsidy to the student in that these loans require much lower interest rates than private loans because the government has the power to ensure repayment, thereby virtually eliminating the risk. It could reasonably be argued that this subsidy benefits the community by a eliminating a moral hazard externality in the human capital market.
The proponents of free college and university education may then argue that the problem with this government loan policy is that many individuals who thereby become educated will find the future payments too high relative to the income they will subsequently be able to earn, with the result that they gain nothing from their education. In most cases where the expected future returns from the education are insufficient to cover its cost, it is reasonable to conclude that the investment is not socially profitable and therefore should not be undertaken. Nevertheless, it should be kept in mind that there may be some fields of study---such as, for example, classics and ancient history---in which there are very few employment opportunities in the private sector. It may thus be necessary to subsidize the acquisition of human capital in these areas to satisfy a positive externality taking the form of benefits from preserving and increasing human knowledge. Of course, the required subsidy could simply take the form of public employment at appropriate salaries for a sufficient number of individuals to eliminate the externality.