They insist that we should observe the Sabbath as Jesus Christ did, because He is our example par excellence. They are correct, but not in the manner they suppose.
Jesus Christ is indeed our Example par excellence. Paul the apostle could therefore admonish us to "Imitate me, just as I also imitate Christ" (1 Corinthians 11:1).
However, we are not to follow the example of Jesus Christ with regard to the Sabbath of the law. That law was the "law of Moses" (Luke 2:22). It was summarized in the Ten Commandments (Exodus 20:1-17), and expanded in "The Judgments" (Exodus 21ff; et.al.). It included both moral and ceremonial commandments. The moral commandments are those which God has ever required of every man (e.g. commandments 1-3 & 5-10 of the Ten Commandments; cp. Matthew 22:37-40 & Mark 12:30f). The ceremonial commandments were those observances and rituals which God required only of Israel in the covenant He made with them at Mount Sinai.
When the Son of God assumed humanity, having been "born of a woman," He was also "born under the law" (Galatians 4:4). He therefore faithfully obeyed not only all its moral commandments, but also all its ceremonial commandments.
For example: 1) He was circumcised and presented in the temple (Luke 2:21-24) in obedience to Leviticus 12:2-6. 2) He paid the temple tax (Matthew 17:24-27) in obedience to Exodus 30:13 and 38:26. 3) He commanded healed lepers to report to their priests (Luke 17:11-14) in obedience to Leviticus 14:2ff. 4) He observed the feasts and holy days of the Old Covenant (e.g. Matthew 26:18) in obedience to Leviticus 23:4-14. And 5) He observed the Sabbath of the fourth commandment.
But it would be contrary to the gospel for men today to follow the example of Jesus Christ in observing these ceremonies of the law. Circumcision and temple taxes and reporting to priests and holy days and sabbaths belonged to a covenant and law no longer in effect, not to the gospel (see e.g. Romans 2:25-29; Philippians 3:1-9). God requires His people today to "Therefore let no one judge you ... regarding ... sabbaths ...." (Colossians 2:16f). God's people today are therefore not required to follow the example of Jesus Christ with regard to the Sabbath of the law.
Rather, we are to follow the example of Jesus Christ with regard to the Sabbath of the gospel. As God entered into an everlasting Sabbath upon completing the work of creation (Genesis 2:1-3), Jesus Christ has entered into an everlasting Sabbath upon completing the work of salvation. After shouting just before He died, "It is finished!" (John 19:30), He entered into His everlasting rest at the right hand of God.
And we are exhorted to follow the example of Jesus Christ in entering into this gospel Sabbath. "There remains therefore a rest [Greek sabbatismos, a "sabbath-keeping"] for the people of God. For he who has entered His rest has himself also ceased from his works as God did from His. Let us therefore be diligent to enter that rest, lest anyone fall after the same example of disobedience" (Hebrews 4:9-11). This Sabbath is not that of the law, in which the Hebrews ceased from physical works. Rather, it is the Sabbath of salvation, into which we enter by faith in Jesus Christ, and in which we cease from works of self-righteousness, reposing forevermore in only the finished work of salvation wrought by Jesus Christ.
Have you entered into this gospel Sabbath?