The Book of Mormon follows the Bible in regards to the Sabbath, and is associated with the Law of Moses:
5 And now, behold, two hundred years had passed away, and the people of Nephi had waxed strong in the land. They observed to keep the law of Moses and the sabbath day holy unto the Lord. And they profaned not; neither did they blaspheme. And the laws of the land were exceedingly strict. (Jarom 1:5)
The prophet Abinadi recited the Ten Commandments to the wicked king Noah and his priests:
16 Remember the sabbath day, to keep it holy.
17 Six days shalt thou labor, and do all thy work;
18 But the seventh day, the sabbath of the Lord thy God, thou shalt not do any work, thou, nor thy son, nor thy daughter, thy man-servant, nor thy maid-servant, nor thy cattle, nor thy stranger that is within thy gates;
19 For in six days the Lord made heaven and earth, and the sea, and all that in them is; wherefore the Lord blessed the sabbath day, and hallowed it. (Mosiah 13:16-19)
The point is not which day of the week it should occur on, but that a person apply the principle of one in seven.
When Jesus visited the people of The Book of Mormon after his resurrection, he said nothing regarding the Sabbath Day. He did however say that sacrifices and outward offerings should be replaced with the offerings of a contrite spirit and a broken heart:
18 I am the light and the life of the world. I am Alpha and Omega, the beginning and the end.
19 And ye shall offer up unto me no more the shedding of blood; yea, your sacrifices and your burnt offerings shall be done away, for I will accept none of your sacrifices and your burnt offerings.
20 And ye shall offer for a sacrifice unto me a broken heart and a contrite spirit. And whoso cometh unto me with a broken heart and a contrite spirit, him will I baptize with fire and with the Holy Ghost. (3 Nephi 9:18-20)
It is unfounded to claim that ones salvation is based upon whether they keep a sabbath day; or upon which day of the week they keep it. The Book of Mormon is very clear what the requirements of salvation are, and this is not one of them:
33 Therefore, keep these sayings which I have commanded you that ye come not under condemnation; for wo unto him whom the Father condemneth.
34 And I give you these commandments because of the disputations which have been among you. And blessed are ye if ye have no disputations among you. (3 Nephi 18:33-34.)
The doctrine Jesus taught was purposeful and specific. What was the purpose? To stop disputations! Those looking for something to judge others by should remember what Paul said:
Let no man therefore judge you in meat, or in drink, or in respect of an holyday, or of the new moon, or of the sabbath days: (Colossians 2:16.)