The Lanarkshire side go into the crunch double-header on a high following their second place finish in the Championship and the 2-1 aggregate semi-final win over Falkirk.
Tony Andreu, who scored in Accies' 1-0 second-leg victory over the Bairns on Sunday, said afterwards that he did not fear the Easter Road side while Hamilton player/manager Alex Neil claimed he would not like to be in Butcher's shoes given that his side won only one of their last 18 league games.
However, the Hibs manager said: "They can say what they want and they are certainly doing that. We just get on with our job.
"We have had a lot of criticism this year and rightly so because we haven't played anywhere near the level we can do and we are in a position where we didn't want to be.
"You can talk and say things and Hamilton are coming into this match very confident because they won their last league game by 10 goals (against Morton) and had a great season.
"They are certainly on a high and there is a lot of talk, that they have positivity and we have negativity but I don't see negativity in our camp.
"I see only positive signs and good signs and a group of players who are determined to do well and keep us in the Premiership."
Neil acknowledged Hibs' Premiership pedigree and admitted that the Leithers' recent form, if not their results, had been good and Butcher senses that the Easter Road side are about to benefit from a change in fortune.
The former Motherwell and Inverness boss said: "Bad runs and good runs have to come to an end sometime.
"If we keep on playing the way we have been playing and making the chances that we have been making, then we will be fine.
"You can look back and say we didn't do this and we didn't do that, it is what we will do that concerns me and I am looking forward to seeing these boys in the two games.
"It is a big game at Hamilton, there is going to be a full house with lots of Hibs fans there and a cracking atmosphere.
"These are games where you want big players to stand up and we have big, strong players who know they have to stand up and be counted because it is very much crunch time."
Veteran defender Alan Maybury admits he does not want a relegation spoiling his C.V.
The 35-year-old Irishman said: "I have never come so close to relegation or played in these games.
"I think it is a big thing. There is a lot riding on it.
"But it is another opportunity try to get the club out of the mess we got ourselves into.
"You can talk yourself into a black hole if you don't get your head up, stick your chest out, be positive and be confident in the group.
"You have to care. There should be a pride about your work whatever it is, showing professionalism and a standard of work is a minimum requirement and I think the players here have that."