Shabbat and Jewish holidays should not be observed by the Bnei Noach. Why?
The Torah says in Genesis 8:22, “Day and night they shall not rest.” From this, our sages in the Talmud (Sanhedrin 58b) learn that “a Gentile who rests is liable to the death penalty.”.
The concept of “rest” here refers to taking a day of the week as a religious day of rest, that is, in the name of God, even if it is not the seventh day of the week. The concept of “death” here refers to heavenly death., not to death by an earthly court.
This idea is mentioned in Jewish law (halakha) in Rambam, Laws of Kings, chapter 10, law 9. This means that Gentiles are forbidden from observing Shabbat. This does not mean that they cannot rest in the simple sense of the word, but rather that such rest cannot be in honor of God creating the universe in six days and resting on the seventh.
In turn, in Exodus 31:12-17 the Torah clearly states that Shabbat is a day of celebration exclusively for the Jewish people.
12. God said to Moses
13. that he should speak to the Israelites and say to them: You must still keep My Sabbaths. It is a sign between Me and you for all generations, so that you may understand that I, God, am making you holy.
14. Therefore, observe the Sabbath as something sacred to you. Anyone who does work [on the Sabbath] will be spiritually cut off from his people, and therefore, anyone who violates it will be condemned to death.
15. Do your work six days a week, but keep the Sabbath day holy, the Sabbath of Sabbaths. Anyone who does any work on the Sabbath day must be put to death.
16. The Israelites, in this way, will keep the Sabbath, making it a day of rest for all generations, as an everlasting covenant.
17. It is a sign between Me and the Israelites that during the six days of the week God made heaven and earth, but on the Sabbath He ceased to work and withdrew to the spiritual.
Rashi, one of the leading commentators on the Torah, explains regarding the quoted verse 13 that the idea of "a sign" is that it is a demonstration of the greatness of the Jewish people that God bequeathed them the Sabbath. He then explains that "to make them understand" refers to all the nations of the world knowing that "I, God, am making them holy," meaning that God sanctifies the Jewish people with the Sabbath.
Likewise, all the festivals mentioned in the Torah are signs of the bond between the Jewish people and God, and the same law applies so that the Gentiles should not observe those festivals.
The exceptions to this rule are the holidays of Rosh Hashanah, when we commemorate the creation of Man, all human beings, and Yom Kippur, the Day of Atonement.
The Talmud (Rosh Hashanah 16a) explains that Rosh Hashanah is the Day of Divine Judgment for all humankind, and Yom Kippur, the Day of Atonement, is the day on which that judgment is sealed. From this, it follows that these two Jewish holidays also have a connection to Gentiles.
However, this relationship is not with the specific precepts of each of these holidays, such as hearing the sound of the Shofar on Rosh Hashanah and fasting on Yom Kippur, but with the conceptual content of the holiday: Rosh Hashanah as a day of judgment and acceptance of God as King over all creation and Yom Kippur as a day of atonement for transgressions and Divine forgiveness.
The proper way for a Gentile to spend these days is to recite psalms. All the psalms. On Yom Kippur, one may add, at some point during the day, a confession of the sins of the previous year, and even of years before that. But not as obligation.