The midfielder was ordered off in the 36th minute after a tussle with Dylan McGeouch then Anthony Stokes.
It followed a flare-up right in front of referee John Beaton which also resulted in bookings for Rangers' Graham Dorrans and McGeouch.
"We are going to appeal it, and we will show the images," said Caixinha.
"I invite the referee to watch it, and to be honest, normally when someone commits a mistake, at least to my principles, I would recognise it."
Jack's dismissal came with the scores locked at 1-1 after Hibs striker Simon Murray had cancelled out Alfredo Morelos' third-minute opener.
James Tavernier's own goal put Hibs in front, and though he would make amends by scoring at the right end nine minutes from time, Vykintas Slivka had already scored an impressive third for the visitors.
Caixinha told Rangers TV: "I am not the type of guy which speaks about referees and I always take my responsibilities, but I think everyone involved in the game needs to take their responsibilities.
"I know I am going to take care of my boys and I know the character of the boys I have down there, and I know they will have a strong reaction.
"This game is not about only playing for yourself - there are a lot of elements which we also need to improve. But we are here, we are the people and we will keep going in front."
The Portuguese also appeared to suggest that Hibs manager Neil Lennon and his assistant Garry Parker were treated differently by fourth official Euan Anderson.
He said: "I am not talking about influence, I am talking about having the same treatment.
"If the opponents are allowed to have two coaches in the technical area, we are also allowed to have it.
"If I listen to what I listened to today, with the other manager getting to the fourth official the way he did, that allows me to go the same way but I don't have the same principles."
Manager Pedro Caixinha has confirmed Rangers will appeal against the red card shown to Ryan Jack in the 3-2 defeat to Hibernian.