It's just a Gregor thing
I'll reiterate my point from earlier, George R. R. Martin has written Gregor Clegane as a "murderous brute":
He's a murderous brute, and really needs no reason to kill someone.
-So Spake Martin http://www.westeros.org/Citadel/SSM/Entry/Asshai.com_Interview_in_Barcelona.
Imagine how much more murderous and brutish he is once revived as an undead version of himself. GRRM has also said about those who survive death that:
My characters who come back from death are worse for wear. In some ways, they're not even the same characters anymore. The body may be moving, but some aspect of the spirit is changed or transformed, and they've lost something. One of the characters who has come back repeatedly from death is Beric Dondarrion, The Lightning Lord. Each time he's revived he loses a little more of himself. He was sent on a mission before his first death. He was sent on a mission to do something, and it's like, that's what he's clinging to. He's forgetting other things, he's forgetting who he is, or where he lived. He's forgotten the woman who he was once supposed to marry. Bits of his humanity are lost every time he comes back from death; he remembers that mission. His flesh is falling away from him, but this one thing, this purpose that he had is part of what's animating him and bringing him back to death. I think you see echoes of that with some of the other characters who have come back from death.
This is echoed multiple times with Gregor, given how dead he was and how focused he was on his rage against his brother, Sandor.
You'll recall that this isn't the first time that he's disobeyed orders and protocol to go after Sandor, in Season 1 of Game of Thrones (and the related first book, A Song of Ice and Fire: A Game of Thrones), he attacks and nearly kills no other than Sandor Clegane, during the tourney of the Hand.
Their rivalry has been built up for a very long time, both in-universe, as well as out-of-universe.