Attempted Stoning Indicates More than a Claim to Unity of Purpose or Will
In John 10:30, what did the Lord Jesus Christ mean when he said, "I and my Father are one"?
Perhaps the Lord Jesus Christ meant "I and my Father are one in purpose" or "I and my Father are one in will," but then, how does one explain the Jews' reaction after they heard his statement?
31 Then the Jews took up stones again to stone him.
Certainly such a reaction by the Jews for the Lord Jesus Christ expressing unity of purpose or will was not warranted, but more importantly, the attempt to stone him if it were for a claim of unity of purpose or will was unprecedented.
In John 8:58, it is written,
58 Jesus said to them, "Amen, amen, I say to you, before Abraham was born, I exist."
To which the Jews responded,
59 Then they took stones to throw on him.
Thus, we see that the Jews had previously attempted to stone the Lord Jesus Christ, but was it because he expressed unity of purpose or will? It is evident in John 8:58 that the Lord Jesus Christ was affirming that he existed before Abraham,1 regardless if one believes «ἐγὼ εἰμί» is a reiteration of a divine name (i.e., "I AM"). Because the Jews did not believe the statement of the Lord Jesus Christ, that is, they rejected the notion that he existed before Abraham was born, they considered it blasphemous and picked up stones to stone him. But, they did not attempt to stone him because he expressed a unity of purpose or will in John 8:58.
The second attempted stoning occurrs in John 10:30 after the Lord Jesus Christ states, "I and my Father are one." Not unlike John 8:58-59, the Jews' attempt to stone the Lord Jesus Christ in John 10:30-31 indicates that they considered his statement to be blasphemous. Hence, the statement must have been more than a declaration of unity of will or purpose. To deny this, one must suggest that the Jews attempted to stone the Lord Jesus Christ for expressing a unity of purpose or will (a bizarre and unfathomable suggestion!).
The Explanation of the Unity (Oneness) as the Impetus for Stoning
As discussed, the Jews attempted to stone the Lord Jesus Christ because they considered his statement to be blasphemous. Therefore, rather than unity of purpose or will, which would not have warranted such a reaction, it is more likely that the Lord Jesus Christ, who existed before Abraham was born, was stating that he and the Father were of one nature.
Recall elsewhere that the Lord Jesus Christ distinguishes the manner in which God is his father versus the manner in which God is the father of other people. For example, in John 5:18, the author wrote that Jesus "also said that God was his own (ἴδιον) Father, making himself equal with God." Elsewhere, in John 20:17, the Lord Jesus Christ states, "I ascend to my Father and your Father," not "I ascend to our Father."
Because a father and son share the same nature, and the Lord Jesus Christ was distinguishing his relationship with "his own" Father versus that of other humans with the Father, and the author states that the Lord Jesus Christ made himself equal to God, then it is most likely that the unity declared by the Lord Jesus Christ in John 10:30 is unity of nature. Of course, this unity also implies a unity of purpose and will, but such unity is an effect of the unity of nature. It is the claim to unity of nature that caused the Jews' reaction, not a supposed claim to unity of purpose or will.
In his commentary on John 10:30, Hermann Olshausen wrote,

Footnotes
1 "Before Abraham was born, I exist."
References
Olshausen, Hermann. Biblical Commentary on the New Testament. Trans. Kendrick, A. C. Vol. 2. New York: Sheldon, 1866.