Your question depends on three ingredients: a sound definition of consciousness, a sound philosophical anthropology, and a sound definition of "computer". Given those, one can begin answering questions such as
- why do human beings experience existential angst?
- what is the relationship between computer and human being?
- is consciousness necessary to have an encounter with existential questions?
- can computers experience existential angst or at least run into existential questions?
- can computers be conscious?
- what is the relationship between reason/understanding and consciousness?
These aren't easy questions, and neither is yours, IMO. And since this is an interdisciplinary question, one would need to reconcile the various languages used in each domain. My cursory answer flow from some of my positions or tendencies of which I am aware. I avoid the common tendency to anthropomorphize computers or the wild and presumptuous oversimplifications of reductionism.
One source of this angst may be the awareness of one's radical freedom. Another source may be the futility of one's actions as death renders one and everything naught. In the case of freedom, if we accept that humans beings are free or capable of being free, it is clear that computers are not free in this sense, as they are fully deterministic machines able only to do what they are programmed to do, whether directly or indirectly. Their actions are rooted not in a finality of their own, but the finality of their human creators and programmers. In another sense, they are free as long as they are not impeded in executing instructions, but this is not the kind of freedom human beings experience. In second case (the futility of all action), this again cannot escape teleology if it is to make rational sense. A futile action is one which brings no (lasting) benefit, which itself requires that some desire be the motive force. Computers do not desire; they only execute instructions. Since computers remain idle without instructions put into it, then their nature is to execute instructions which precludes autonomousness. One can program a computer to perform certain actions once certain conditions are met, but I am far from convinced that the lead from computer to human being can be so easily made. It has also never been shown that a computer can reason.
Now since a computer does not desire, it follows that even if a computer could come to hold some representation of such a preposition that could produce a condition wherein the categorical futility of action could alter the course of instructions, in the best case it would only cease to act. But a computer must be first programmed in such a way that it would act on such a preposition in such a way that it would cease acting i.e. executing instructions (we could equally program it to play chess). The meaning of such prepositions is lost on the computer. It does not act by virtue of the meaning, but by virtue of the programming.
Enter consciousness. Does consciousness entail autonomy, understanding, meaning or reason? That's affording a computer quite a bit, and its not all that clear what they could mean or how they could exist outside of the human context. Regardless, one must ask "what could a computer be conscious of?" and "how is it relevant to this problem?".