Answer: Yes. There are other verses that refute this eisegesis.
The JOINING of two groups in Eph 2:14-16 were prophesied about in old covenant scripture. So let's see what was prophesied:
Who did Jeremiah say would be joined?
Jer 3:17-18 - At that time Jerusalem shall be called the throne of the Lord, and all nations shall gather to it, to the presence of the Lord in Jerusalem, and they shall no more stubbornly follow their own evil heart. In those days the house of Judah shall JOIN the house of Israel, and together they shall come from the land of the north to the land that I gave your fathers for a heritage.
Who did Ezekiel prophesy would be joined?
Eze 37:15-17 - The word of the Lord came to me: “Son of man, take a stick and write on it, ‘For Judah, and the people of Israel associated with him’; then take another stick and write on it, ‘For Joseph (the stick of Ephraim) and all the house of Israel associated with him.’ And JOIN them one to another into ONE stick, that they may become ONE in your hand.
and again:
Eze 37:19 - Thus says the Lord God: Behold, I am about to take the stick of Joseph (that is in the hand of Ephraim) and the tribes of Israel associated with him, and I will JOIN with it the stick of Judah, and make them ONE stick, that they may be ONE in my hand.
- Who did Hosea say would be joined?
Hos 1:10-11 - And in the place where it was said to them, “You are not my people,” it shall be said to them, “Children of the living God.” And the children of Judah and the children of Israel shall be gathered TOGETHER, and they shall appoint for themselves ONE head. And they shall go up from the land, for great shall be the day of Jezreel.
And again:
Hos 2:23 - I will have mercy on No Mercy, and I will say to Not My People,‘You are my people’; and he shall say, ‘You are my God.’”
So who was No Mercy?
Hos 1:6 - I will no more have mercy on the house of Israel
Who was Not My People?
Hos 1:9 - {to the house of Judah} And the Lord said, “Call his name Not My People, for you are not my people, and I am not your God. {see the next verse}
Clearly the old covenant prophets thought God was going to join ONE group to another {from the above verses}. Was it thought to be the two groups mentioned above in the Question?
So then how did Yehshua's contemporaries understand it?
Let's see:
In John 7:33 "Yehshua then said, “I will be with you a little longer, and then I am going to him who sent me."
When Yehshua spoke of his departure where did the listeners assume Yehshua would leave to?
John 7:35 - The Jews said to one another, “Where does this man intend to go that we will not find him? Does he intend to go to the exiles {of the house of Israel} among the Greeks and teach the Greeks? (2 Kings 17:23,1 Chr 5:26, Amos 7:11 et. al.)
Who did Yehshua say he was sent only to?
Matt 15:24 - He answered, “I was sent only to the lost sheep of the house of Israel.”
Where did Yehshua instruct his servants to go?
Matt 10:5-6 - "Go nowhere among the Gentiles and enter no town of the Samaritans, but go rather to the lost sheep of the house of Israel.
When Yehshua commanded his followers to wait in Jerusalem for the arrival of the Holy Spirit, what other group alienated from the Commonwealth of Israel , did they express concern for?
Acts 1:6 - When they had come together, they asked him, “Lord, will you at this time restore the kingdom to {the house of} Israel?”
Where were the exiles of the house of Israel, geographically?
1 Peter 1:1 - To those who are elect exiles of the dispersion in Pontus, Galatia, Cappadocia, Asia, and Bithynia ...
... and so on.
Yehshua's contemporaries, indeed Yehshua himself, were also concerned about another sheepfold (Mic 2:12). Apparently they were concerned about the same sheepfold the prophets of old were.
Of course, we could simply ignore the plain meaning of the many scriptures just cited, all of which clearly suggest some other interpretation than the one reflected in the question above. We could pretend they don't exist, or read them less plainly. Instead we could read into John 10:16 some other interpretation more pleasing to our bias (and tradition).