Chaim looked at the kippa resting on his palm. True, a few dollars payed to any Judaica store would result in a much more finished look. However this kippa, Chaim’s kippa, which he had made with his own two hands, here in the darkness of Siberia, meant so much more. As he placed the rough fabric on his head, he felt a sense of elation. The kippa was his way of conveying to the hardened prison guards, and, more importantly, to himself, a vitally important message. They had taken him from his home, his family, his community. There was one thing they couldn’t take away, though. Never, never would they be able to take away his connection to G-d.
It wasn’t long before a guard took notice of Chaim’s new headwear. And its significance did not escape the guard’s notice.
"Jew! Take that off your head or die!" the guard sneered.
The gun aimed in Chaim’s direction made him flinch for a second, but Chaim quickly recovered his composure. He stood straighter and looked the guard in the eye, but said nothing.
"Jew, did you not hear me the first time? Don’t you know what a gun is?!" The guard was now yelling.
Chaim stood stoically, waiting.
"Are you not afraid to die?" gasped the guard.
Slowly, calmly, Chaim spoke. "My kippa, this piece of cloth on my head, means that I know there is someone above me. He and He alone determines whether I live or die. So, no, I am not afraid to die. You believe that life and death are in the hands of Brezhnev, so you are terrified of death."
Was Chaim allowed to endanger his life, in order to continue wearing his kippa?

Answer of Rabbi Yaakov Ariel, shlita:
If a Jew is given the choice to commit idolatry, sexual immorality or murder, or be killed, the Jew must choose to give up his life rather than violate any of those three sins. However, if it is a matter of committing any other sin, or being killed, the Jew should commit the sin, and prolong his life. These rules apply under general circumstances. However, at a time when Jews are being pursued, in an attempt to convert Jews to a foreign religion, a Jew must give up his life rather than violate any prohibition, even a relatively minor commandment. (Sanhedrin 74, Shulchan Aruch 157, 1)
In Russia, at the time of this story, Jews were being actively pursued, in order to draw them away from Judaism. Therefore, Chaim was correct in refusing to remove his kippa.
It is important to note that, if the guard would, himself, have forcibly removed the kippa from Chaim’s head, Chaim would have been allowed to remove his kippa on his own. This is because Chaim would be incapable of observing the mitzva regardless, so it would make no difference if the kippa was removed by the guard or by Chaim. (See Biur HaGra on the above mentioned Shulchan Aruch, note 10.)
The authorities debate whether a Jew is allowed to give up his life under circumstances in which he is not actually required to give up his life. According to Maimonides, it would be forbidden for the person to give up his life, but according to others, it would be permitted. (See the commentary of the Shach on the above mentioned Shulchan Aruch, se’if katan 10.) In practice, many Jews throughout the generations chose to give up their lives, even in situations in which they were not obligated to do so.
In summary: Because our story took place at a time when the Russians were actively seeking to prevent Jews from keeping the Torah, Chaim acted correctly in choosing to give up his life, rather than remove his kippa.