The general consensus from epidemiological game-theory studies is that vaccination coverage driven by self-interest (Nash vaccination) is generally lower than group-optimal coverage (utilitarian vaccination). However, diseases that become more severe with age, such as chickenpox, pose an exception to this general consensus. An individual choice to be vaccinated against chickenpox has the potential to harm those not vaccinated by increasing the average age at infection and thus the severity of infection as well as those already vaccinated by increasing the probability of breakthrough infection. To investigate the effects of these externalities on the relationship between Nash and utilitarian vaccination coverages for chickenpox, we developed a game-theory epidemic model that we apply to the USA and Israel, which has different vaccination programmes, vaccination and treatment costs, as well as vaccination coverage levels. We find that the increase in chickenpox severity with age can reverse the typical relationship between utilitarian and Nash vaccination coverages in both the USA and Israel. Our model suggests that to obtain herd immunity of chickenpox vaccination, subsidies or external regulation should be used if vaccination costs are high. By contrast, for low vaccination costs, improving awareness of the vaccine and the potential cost of chickenpox infection is crucial.