I originally thought there was no way anyone had studied this - it felt like one of those questions we get here a lot where people seem to assume "scientists know (almost) everything" - but I decided to give a check anyways.
There is in fact evidence that some insects could be scared to death. However, the process is a bit less dramatic than the one you suggest.
I didn't see much on mosquitoes specifically, but your question seems to allow extrapolation from other insects.
I found at least one paper, McCauley, et al. 2011, that studied whether the mere presence of a predator could cause fatal reactions, in this case in dragonflies. There are several others that show more generic stress-induced responses.
McCauley, et al. exposed dragonfly larvae to predators (either fish or a larger dragonfly) who were unable to actually eat the larvae because the larvae were contained within a 'cage' within a larger aquarium.
The larvae exposed to either predator type had substantially more mortality, and only about 30% as many survived to adulthood. This is not precisely insects being startled to death, but rather they propose that the chronic stress of predator exposure caused their growth to be impaired. An alternative explanation is that the larvae exposed to predators simply spent too little time eating to obtain enough nutrition, but they argue in their discussion that this is not likely to be the only cause in the case of the dragonfly predators, because unlike the fish those predators did not reduce their foraging behavior.
There are many other papers that talk about "non-consumptive mortality" more generally but these often include causes of death such as succumbing to injury from failed/incomplete predation, energetic expenditures, etc (Siepielski et al., 2014; Veselý et al., 2017) - few others actually use a highly controlled environment to prevent physical contact between predator and prey like in McCauley. Other studies do show that exposure to increased stress levels causes changes in gene expression and impacts on the immune system in insects (De Block & Stoks, 2008; Slos & Stoks, 2008).
On the point of mosquitoes, I did find one paper that showed reduced lifespan duration in adult mosquitoes exposed to increased predation as larvae (Bellamy & Alto, 2018) - again not exactly what you were looking for in terms of an acute event, but still somewhat related.
I think an important takeaway is that insects, like vertebrates, can be susceptible to stress.
Bellamy, S. K., & Alto, B. W. (2018). Mosquito responses to trait-and density-mediated interactions of predation. Oecologia, 1-11.
De Block, M., & Stoks, R. (2008). Compensatory growth and oxidative stress in a damselfly. Proceedings of the Royal Society of London B: Biological Sciences, 275(1636), 781-785.
McCauley, S. J., Rowe, L., & Fortin, M. J. (2011). The deadly effects of “nonlethal” predators. Ecology, 92(11), 2043-2048.
Siepielski, A. M., Wang, J., & Prince, G. (2014). NONCONSUMPTIVE PREDATOR‐DRIVEN MORTALITY CAUSES NATURAL SELECTION ON PREY. Evolution, 68(3), 696-704.
Slos, S., & Stoks, R. (2008). Predation risk induces stress proteins and reduces antioxidant defense. Functional Ecology, 22(4), 637-642.
Veselý, L., Boukal, D. S., Buřič, M., Kozák, P., Kouba, A., & Sentis, A. (2017). Effects of prey density, temperature and predator diversity on nonconsumptive predator-driven mortality in a freshwater food web. Scientific reports, 7(1), 18075.