Short answer
No. Jesus says "I and my Father are one" while speaking of his sheep, meaning his followers. No one can snatch them from his hand, and no one can snatch them from the Father's hand. In the verse before, he calls the Father "greater than all". So the oneness is of purpose, not of being. In John 17 he prays that all believers "may be one, even as we are one". That cannot mean every believer is God. When the crowd said he made himself God, he pointed to a psalm that calls men "gods". What he had claimed, he said, was "I am the Son of God". Later he says, "my Father is greater than I". Islam's answer is the one Jesus gives in the Qur'an: "It is Allah Who is my Lord and your Lord".
Full explanation
Read the verses around it
Jesus is talking about his sheep, meaning his followers. He says no one can pluck them out of his hand. He says no one can pluck them out of the Father's hand either. In the same breath he says the Father, who gave them to him, is greater than all. So the two are one in keeping the sheep safe, and the Father is the greater.
“And I give unto them eternal life; and they shall never perish, neither shall any man pluck them out of my hand.”
“My Father, which gave them me, is greater than all; and no man is able to pluck them out of my Father’s hand.”
“I and my Father are one.”
The same "one" for every believer
In John 17 Jesus prays for his disciples and for everyone who will believe through them. He asks that they may be one as he and the Father are one. The Greek word for "one" there is hen, the same word as in John 10:30. Paul, a Christian missionary of the first century, uses hen too, for the one who plants and the one who waters. Jesus also speaks of being in his followers: "I in them, and thou in me". If the word made Jesus God, it would make every believer God.
“Neither pray I for these alone, but for them also which shall believe on me through their word;”
“That they all may be one; as thou, Father, art in me, and I in thee, that they also may be one in us: that the world may believe that thou hast sent me.”
“And the glory which thou gavest me I have given them; that they may be one, even as we are one:”
“I in them, and thou in me, that they may be made perfect in one; and that the world may know that thou hast sent me, and hast loved them, as thou hast loved me.”
“Now he that planteth and he that watereth are one: and every man shall receive his own reward according to his own labour.”
One in will, not in being
Elsewhere in John, Jesus explains why God (﷾) is with him. God (﷾) is with him because he always does what pleases Him. He says he can do nothing by himself, and that he seeks the will of the One who sent him. That is a oneness of purpose between a sender and the one he sends.
“And he that sent me is with me: the Father hath not left me alone; for I do always those things that please him.”
“I can of mine own self do nothing: as I hear, I judge: and my judgment is just; because I seek not mine own will, but the will of the Father which hath sent me.”
"My Father is greater than I"
In the same Gospel Jesus says plainly that the Father is greater than he is. If the two were one in being, neither could be greater. Christians say he spoke only as a man. But natures do not speak; persons do. The one "I" who said it puts the Father above himself. Peter, one of his disciples, later preached that Jesus was a man approved of God (﷾), through whom God (﷾) did miracles.
“Ye have heard how I said unto you, I go away, and come again unto you. If ye loved me, ye would rejoice, because I said, I go unto the Father: for my Father is greater than I.”
“Ye men of Israel, hear these words; Jesus of Nazareth, a man approved of God among you by miracles and wonders and signs, which God did by him in the midst of you, as ye yourselves also know:”
The council confessed one and the same Christ in two natures, divine and human, and declared him "not parted or divided into two persons, but one and the same Son".
How Christians read John 10:30
Christians read "one" as one in nature. They say the Son shares the being of the Father while remaining a distinct person. They point out that the crowd took up stones at once. The crowd said it was because he, a man, made himself God.
“I and my Father are one.”
“The Jews answered him, saying, For a good work we stone thee not; but for blasphemy; and because that thou, being a man, makest thyself God.”
His answer to the charge
When the crowd accused him of making himself God, Jesus did not accept the charge. He quoted Psalm 82, a psalm (a Hebrew sacred song) where human judges are called gods and children of the Most High. Then he said that his claim was to be the Son of God.
“The Jews answered him, saying, For a good work we stone thee not; but for blasphemy; and because that thou, being a man, makest thyself God.”
“Jesus answered them, Is it not written in your law, I said, Ye are gods? If he called them gods, unto whom the word of God came, and the scripture cannot be broken; Say ye of him, whom the Father hath sanctified, and sent into the world, Thou blasphemest; because I said, I am the Son of God?”
“I have said, Ye are gods; and all of you are children of the most High.”
What the Qur'an says Jesus taught
The Qur'an records Jesus telling his people that Allah (﷾) is his Lord and their Lord. He tells them to worship Him alone. This is the same message the verses above point to.
“"'It is Allah Who is my Lord and your Lord; then worship Him. This is a Way that is straight.'"”
“They do blaspheme who say: "Allah is Christ the son of Mary." But said Christ: "O Children of Israel! worship Allah, my Lord and your Lord." Whoever joins other gods with Allah,- Allah will forbid him the garden, and the Fire will be his abode. There will for the wrong-doers be no one to help.”
Verse by verse
John 10:30 — Gospel of John, chapter 10, verse 30
“I and my Father are one.”
Cited for
Christians note that the Greek word for "one" (hen) is neuter, "one thing" rather than "one person". They read it as one divine nature shared by two persons.
The response
John uses the same word when Jesus prays that his followers "may be one, even as we are one". Paul uses it of the one who plants and the one who waters. In neither place does it mean one nature.
Where it lands — The word gives believers the same oneness it gives Jesus, so it cannot mean that Jesus is God.
John 10:33 — Gospel of John, chapter 10, verse 33
“The Jews answered him, saying, For a good work we stone thee not; but for blasphemy; and because that thou, being a man, makest thyself God.”
Cited for
The crowd understood Jesus to be claiming to be God, and he did not deny it.
The response
He did answer them. He quoted Psalm 82, where men are called gods, and said his claim was to be the Son of God. That is a correction of their charge, not an admission.
Where it lands — Jesus answered the charge by appealing to the title Son of God, which scripture gives to men, not by claiming to be God.
John 10:38 — Gospel of John, chapter 10, verse 38
“But if I do, though ye believe not me, believe the works: that ye may know, and believe, that the Father is in me, and I in him.”
Cited for
Christians read "the Father is in me, and I in him" as two divine persons being in each other.
The response
Jesus uses the same language of believers. He says "I in them, and thou in me" (John 17:23). He also says "ye in me, and I in you" (John 14:20). He shares this closeness with every believer.
Where it lands — The "in me, and I in him" language is given to all believers, so it does not make Jesus God.
Perspectives
Mainstream Christian
Christians say Jesus and the Father are one in being, yet distinct persons. The Greek word for "one" (hen) is neuter, meaning "one thing" rather than "one person". Christians read it as one nature.
In John 10:30 Jesus says he and the Father are one, which Christians read as oneness of nature.
Rebuttal
Jesus does say "one", and the Greek word means "one thing". But John 17:22 gives the same word to believers, who are not God. In John 10 itself, the oneness is about keeping the sheep safe, and the Father who gave him the sheep is "greater than all".
The crowd understood the words as a claim to be God and took up stones.
Rebuttal
The crowd did accuse him of making himself God. But Jesus answered them. He quoted Psalm 82, where men are called gods, and said what he had claimed was "I am the Son of God".
The Islamic perspective
Jesus was one with God (﷾) in purpose and obedience, as every true prophet is. He never claimed to be God, and he said plainly that the Father is greater than he is.
In context, Jesus and the Father are one in holding the sheep safe, and the Father is "greater than all".
Jesus prays that all believers may be one as he and the Father are one, using the same word.
Jesus was accused of making himself God. He answered from Psalm 82 that what he had said was "I am the Son of God".
In the same Gospel Jesus says, "my Father is greater than I".
Jesus said he always does what pleases the Father, and seeks the will of the One who sent him. This is a oneness of will.
The Qur'an teaches that Jesus told his people that Allah (﷾) is his Lord and theirs.