JOb 19:27; what does he mean by "my eyes shall behold, and not another"?

Sounds very confidential and "hush-hush"  to me.

Verse 26 is part of that.  Job knew that he would see the Lord in person, not somebody else, after his life.

King David specially mention the kidneys? It was in order to make his point. He was magnifying the depth of penetration on the part of God’s spirit and visual powers, reaching as far down as into Sheol, the common grave of mankind.

 God can also peer down deep into the human body, as far down as the kidneys, these being below the heart and the diaphragm and toward the base of the spine.

 And when God created the embryo and kept it screened off in the belly of David’s mother, he could see down to the location for these deep-seated organs.

 Nothing blocked his visual powers or the range of his creative spirit.   God can look inside the body, whereas humans cannot.

Job is talking about how he expects to see God fact to face.  That even if he is dead and his body has rotten away, he will still stand face to face before God in that flesh someday.  And it will not be him "recreated" as some groups (mis)teach, or something "like" him that will be brought back from a "soul sleep".  But it will him in his body face to face with God. 

https://biblehub.com/commentaries/job/19-27.htm