Did Jesus command His Disciples to Worship Him?

This question puzzled me also. The answer is subtle. If you read through the book of John, you will find that:

1. Jesus identified Himself with God. "before Abraham was born, I am.” (8:58) "I and the Father are one." (10:30)

2. His critics understood this and condemned Him for it.

3. His response was not "you misunderstand", but rather, "Here is why you should believe me."
4. The disciples understood this and worshiped Jesus.
5. Jesus did not correct them for doing so. (Paul Chernoch's answer is excellent)

Does Jesus order His disciples to worship Him? No.

Does Jesus authoritatively teach that He is most worthy of worship? Yes.

The answer that was accepted 18 hours after the question posting is not correct, as it overlooks John 10:30 and the entire flow of the book of John.

If you read through the book of John, you will find that: John the Baptist and Jesus gradually reveal to their disciples Jesus's identity. He says he was sent by God, to speak the words of God. He says He came from heaven, to do the works of God. He says the Father has given Jesus status and authority: "He who comes from heaven is above all ... The Father loves the Son and has given all things into His hand." (3:31,35)

Jesus didn't generally acquiesce to demands for sound bite religion. He did, however say, "Follow Me."

TLDR:

1. The Disciples worshiped Jesus on at least two occasions (Matthew 14 & 28).
2. Studying these two stories is the most logical place to look to see if Jesus commanded them to worship him.
3. Their actions were God-worship, not mere respect.
4. The worship was permitted, not forbidden as idolatry (and punished).
5. The context of the scene (words and actions of Jesus and his disciples) indicate at the very least an invitation to worship, or more strongly a command to worship him.

Argument:

The disciples worshiped Jesus, and he did not correct them for doing so:

16 Now the eleven disciples went to Galilee, to the mountain to which Jesus had directed them. 17 And when they saw him they worshiped him, but some doubted. 18 And Jesus came and said to them, “All authority in heaven and on earth has been given to me. 19 Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, 20 teaching them to observe all that I have commanded you. And behold, I am with you always, to the end of the age.” (Matthew 28:16-20)

Barnabas and Paul, after healing a man, refused to accept the people's worship, because they knew they were not divine:

8 Now at Lystra there was a man sitting who could not use his feet. He was crippled from birth and had never walked. 9 He listened to Paul speaking. And Paul, looking intently at him and seeing that he had faith to be made well, 10 said in a loud voice, “Stand upright on your feet.” And he sprang up and began walking. 11 And when the crowds saw what Paul had done, they lifted up their voices, saying in Lycaonian, “The gods have come down to us in the likeness of men!” 12 Barnabas they called Zeus, and Paul, Hermes, because he was the chief speaker. 13 And the priest of Zeus, whose temple was at the entrance to the city, brought oxen and garlands to the gates and wanted to offer sacrifice with the crowds. 14 But when the apostles Barnabas and Paul heard of it, they tore their garments and rushed out into the crowd, crying out, 15 “Men, why are you doing these things? We also are men, of like nature with you, and we bring you good news, that you should turn from these vain things to a living God, who made the heaven and the earth and the sea and all that is in them." (Acts 14:8-15)

The Apostle John tried to worship an angel, but was rebuked for doing so and told to worship God alone:

8 I, John, am the one who heard and saw these things. And when I heard and saw them, I fell down to worship at the feet of the angel who showed them to me, 9 but he said to me, “You must not do that! I am a fellow servant with you and your brothers the prophets, and with those who keep the words of this book. Worship God.” (Revelation 22:8-9)

King Herod received the worship of his subjects and did not stop them from doing so, and so the Lord struck him down because of it:

21 On an appointed day Herod put on his royal robes, took his seat upon the throne, and delivered an oration to them. 22 And the people were shouting, “The voice of a god, and not of a man!” 23 Immediately an angel of the Lord struck him down, because he did not give God the glory, and he was eaten by worms and breathed his last. (Acts 12:21-23)

So to answer your question, Jesus made it clear by example and comparison with other Scripture that it was proper for him to receive our worship, but not earthly kings, angels or even his own followers. Following his commissioning of his disciples, Jesus was not struck down, but instead lifted up into heaven.

Jesus' statement that "All authority in heaven and on earth has been given to me" is a call to worship by implication, because only God has that authority, and the proper response to an encounter with God is to offer worship.

Additional Support:

Lest there be any confusion as to the nature and degree of the Disciple's worship of Jesus, we have the story of Jesus walking on water:

25 Shortly before dawn Jesus went out to them, walking on the lake. 26 When the disciples saw him walking on the lake, they were terrified. “It’s a ghost,” they said, and cried out in fear.

27 But Jesus immediately said to them: “Take courage! It is I. Don’t be afraid.”

28 “Lord, if it’s you,” Peter replied, “tell me to come to you on the water.”

29 “Come,” he said.

Then Peter got down out of the boat, walked on the water and came toward Jesus. 30 But when he saw the wind, he was afraid and, beginning to sink, cried out, “Lord, save me!”

31 Immediately Jesus reached out his hand and caught him. “You of little faith,” he said, “why did you doubt?”

32 And when they climbed into the boat, the wind died down. 33 Then those who were in the boat worshiped him, saying, “Truly you are the Son of God.” (Matthew 14:25)

In this instance it is unmistakeable that the worship was an admission of divinity; walking on water had long been viewed by many cultures in the ANE as a sign of divinity.

What is worship? It is a two-way communication. We bask in the glow of God's glory and reflect it through our lives. Worship is a process whereby God communicates some of His qualities to us, like a signet ring on wax or a relief on clay. Here Peter asks Jesus to call him, and Jesus commands him, "Come." This command enables Peter to temporarily walk on water. Jesus has communicated part of his divine nature to Peter, closing the loop in this act of worship. Thus Jesus' word to Peter, "Come", was a call to worship.