In response to the question when is one supposed to die when being forced to convert to a different faith

Answer;

Being that the maimondes discusses these halachot at length I will partially quote from what he wrote on the subject.

The Maimonides Yesodei haTorah – Chapter Five rules:

Rregarding the Mitzvois it states “vchai bahem” that one is to live by keeping them and not to die by keeping the Mitzvois and therefore if a gentile forces  a Jew either to transgress a mitzvah or he will kill him one is to transgress the mitzvah for thereby he will live.

Three transgressions

However there are three transgression where one is to die rather than transgress those mitzvois and they are idol worship murder and illicit relationship and actions  which by these transgression one is to die and not g-d forbid transgress the sin

Important condition

When does the above apply? When the gentile desires his own personal benefit and not with the purpose to violate  the mitzvah – for example, he forces a person to build a house or cook food for him on the Sabbath, he rapes a woman, or the like.

However, if his intention is solely to have him violate the mitzvot, [the following rules apply:] If he is alone and there are not ten other Jews present, he should transgress and not sacrifice his life.

However, if he forces him [to transgress] with the intention that he violate [a mitzvah] in the presence of ten Jews, he should sacrifice his life and not transgress. [This applies] even if [the gentile] intended merely that he violate only one of the [Torah’s] mitzvot and although it is not one of three mitzvois mentioned above .

A time of decree against Judaism

in times of a decree – i.e., when a wicked king like Nebuchadnezzar or his like will arise and issue a decree against the Jews to nullify their faith or one of the mitzvot – one should sacrifice one’s life rather than transgress any of the other mitzvot, whether one is compelled [to transgress] amidst ten [Jews] or one is compelled [to transgress merely] amidst gentiles.

When does one receive punishment?

If anyone about whom it is said: “Transgress and do not sacrifice your life,” sacrifices his life and does not transgress, he is held accountable for his life.

Reward

When anyone about whom it is said: “Sacrifice your life and do not transgress,” sacrifices his life and does not transgress, he sanctifies [God’s] name. If he does so in the presence of ten Jews, he sanctifies [God’s] name in public, like Daniel, Chananiah, Mishael, Azariah, and Rabbi Akiva and his colleagues. These are those slain by [the wicked] kingdom, above whom there is no higher level. Concerning them, [Psalms 44:23] states: “For Your sake, we have been slain all day, we are viewed as sheep for the slaughter,” and [Psalms 50:5] states: “Gather unto Me, My pious ones, those who have made a covenant with Me by slaughter.”

What happens when one doesn’t sacrifice his life?

When anyone about whom it is said: “Sacrifice your life and do not transgress,” transgresses instead of sacrificing his life, he desecrates [God’s] name. If he does so in the presence of ten Jews, he desecrates [God’s] name in public, nullifies [the fulfillment of] the positive commandment of the sanctification of [God’s] name, and violates the negative commandment against the desecration of God’s name.

Nevertheless, since he was forced to transgress, he is not [punished by] lashing, and, needless to say, is not executed by the court even if he was forced to slay [a person]. The [punishments of] lashes and execution are administered only to one who transgresses voluntarily, [when the transgression is observed by] witnesses, and [when] a warning [was given]

Escape

One who could, however, escape and flee from under the power of a wicked king and fails to do so is like a dog who returns [to lick] his vomit. He is considered as one who worships false gods willingly. He will be prevented from reaching the world to come and will descend to the lowest levels of Gehinnom

In conclusion

With all this in mind accepting another religion is most like avoda zara (example Christianity) and even if isn’t idol worship however if its being done to cause one to leave g-d then in front of ten one is to die, and if its in front of less people yet it’s a time of a decree (example Spanish inquisition or Russian communism) one is not to transgress but rather die “al kiddush hashem”