May a Christian observe the Lord’s Supper alone?

The most specific instructions in Scripture concerning the way the Church should observe the communion meal is found in Paul's teaching in 1Corinthians 11.

1Cor. 11:18 For, in the first place, when you come together as a church, I hear that divisions exist among you; and in part I believe it.
1Cor. 11:19 For there must also be factions among you, so that those who are approved may become evident among you.
1Cor. 11:20 Therefore when you meet together, it is not to eat the Lord’s Supper,
1Cor. 11:21 for in your eating each one takes his own supper first; and one is hungry and another is drunk.
1Cor. 11:22 What! Do you not have houses in which to eat and drink? Or do you despise the church of God and shame those who have nothing? What shall I say to you? Shall I praise you? In this I will not praise you.
1Cor. 11:23  For I received from the Lord that which I also delivered to you, that the Lord Jesus in the night in which He was betrayed took bread;
1Cor. 11:24 and when He had given thanks, He broke it and said, “This is My body, which is for you; do this in remembrance of Me.”
1Cor. 11:25 In the same way He took the cup also after supper, saying, “This cup is the new covenant in My blood; do this, as often as you drink it, in remembrance of Me.”
1Cor. 11:26 For as often as you eat this bread and drink the cup, you proclaim the Lord’s death until He comes.

In this passage, Paul clearly expected the Church to observe this communion meal in the context of a gathering. Paul's instructions include statements like "...when you come together as a Church..." and "...when you meet together..." which strongly suggests Paul expected the meal to be conducted in a corporate setting. 

Furthermore, the original Last Supper observance by Christ was based on the Passover celebration, which was required to be conducted in a group setting:

Ex. 12:3 “Speak to all the congregation of Israel, saying, ‘On the tenth of this month they are each one to take a lamb for themselves, according to their fathers’ households, a lamb for each household.
Ex. 12:4 ‘Now if the household is too small for a lamb, then he and his neighbor nearest to his house are to take one according to the number of persons in them; according to what each man should eat, you are to divide the lamb.

Notice that smaller families were required to join together with to form larger groups that would ensure the lamb was fully consumed. Therefore, it was not possible for a Jew to observe the Passover meal alone. Here again, this pattern argues against an individual observance of the Lord's Supper, since it is modeled after the Passover. (For more background on the connection between Passover the and Lord's Supper, please listen to Lesson 13 of our John's study.)

Finally, observing the Lord’s Supper individually is not necessary. The purpose of the observation, according to Paul, is to witness to the sacrifice of Jesus and His promise to return for the Church (see 1Cor 11:26). By definition, we cannot offer a witness without an audience, so communion's chief purpose according to Paul cannot be achieved when we're alone. 

For all these reasons, we do not believe a Christian should attempt to conduct communion alone. Doing so is likely nothing more than eating a meal.