Though there have always been a number of Christians who have observed the more traditional holidays such as Christmas and Easter (Resurrection Day), there seems to be an increasing number of Baptists who observe the protestant/Roman Catholic celebrations. Part of this problem can be traced back to the Protestant Reformation. Though a lot a good was accomplished, and the doctrine of justification by faith was restored to its proper place, the Reformation itself was incomplete. When Luther, Calvin, Zwingli, and others left Rome, they took a number of traditions with them. These can be seen in their practice of infant baptism, their tendency to mix church and state, and their practice of non-congregational church government. In addition to these, many Protestant churches continue to practice some of the Roman Catholic holidays such as Advent and, as you have questioned, Lent.
Should a Christian observe such a holiday? The Christian needs to base his decision on the basis of the holiday and what the word of God says. First of all, let me deal with secular holidays although you did not refer to them particularly. In the United States we observed such days as Labor Day, Veterans Day, the Fourth of July and the like. Since these do not directly conflict with Scripture, there is no reason the Christian cannot observe them by taking the day off or spending time with family. Lent, however, is a very different religious observance that is steeped in the traditions of Roman Catholicism. Being a former Roman Catholic, I will answer your question from that viewpoint.
Lent, of course, refers to the forty day period between what is called Ash Wednesday and Easter. (The forty days are reminiscent of the forty day period that Jesus spent fasting in the wilderness). During this time, observers of Lent will perform acts of humiliation, fasting, and penitence in preparation for Easter. Typically, participants will “give up” something dear to them (like chocolate, candy, or even alcoholic beverages) during this time. On Ash Wednesday, worshippers receive a spot of ashes on the forehead to remind them of their humanity, the priest saying the words “dust thou art, and to dust thou wilt return.” Some participants may, then, even attend extra church services like the Stations of the Cross followed by the “Adoration of the Blessed Sacrament;” some will say extra prayers. During these observances, participants perform acts of penitence (showing sorrow for one’s sins) and penance (deeds to make amends for sins). By these acts, these people hope to find favour with God. Sincere participants believe their good works will produce merit in His sight. Also, though there are many sincere observers of the Lenten holiday, there are many nominal professors who find nothing more than a religious obligation. Many do not connect Ash Wednesday with Mardi Gras, (the Tuesday before Ash Wednesday). Mardi Gras observers participate in many ungodly and immoral activities all of which are intended to satisfy the flesh. Why do these folks party so much? Because for the next 40 days they will be observing a religious ritual that requires they deny their flesh.
Now the Bible mentions nothing of Lent. Jesus never mentioned it (or any other religious observance like it, including the mass), nor did Paul, Peter, James or John. It is a completely Roman Catholic invention. In reality, the Christian is to live holy and in humility ever day of the year, not just during a special forty day period. Paul says our bodies are to be “a living sacrifice” (Romans 12:1). His body is the “temple of the Holy Ghost” (1Corinthians 6:19), and he is to live holy because his God is holy (1Peter 1:15-16). As for the religious works, the Bible is very plain that “they that are in the flesh cannot please God” (Romans 8:8) and that “no flesh should glory in his presence” (1Corinthians 1:29), so there is no way that the denials of the flesh and good deeds on the part of the participants can merit favour with God. Salvation is completely by faith in the finished work of the Lord Jesus Christ. The believer is saved by grace, not by good works (Ephesians 2:8-9).
Since Lent is a completely Roman Catholic holiday, it would be well for the Christian to steer clear of it. First, as an invention of Roman Catholicism, it has a wrong basis. Second, it honors the flesh. Though they deny flesh, participants are actually bringing honour to “this body of death” in trusting their flesh can bring forth merit to please God. Third, it is deceptive. Those who observe the Lenten rituals actually believe they are meriting favour with God – I was one of them. And, even though no Catholic priest would agree, by Rome’s emphasis on forty days of religious observance (though there are other special periods of religious obligations during the year), the message is given, albeit indirectly, that one can live any way they desire during the year if they are religious during a special church observance. Since the Bible knows nothing of such an observance, the Christian should avoid it. I hope this helps.