The Staggies have gone eight games without a win which has left them sitting second-bottom of the table, their 14-point gap over Hearts helped by the Jambos' 15-point deduction for entering administration.
In the midst of County's struggle sits Butcher's first - and so far only - win since taking over as Hibs boss from Pat Fenlon in November.
When the two sides met in the fourth round of Scottish Cup at the Global Energy Stadium last month, a first-half drive from Danny Handling took the Easter Road side into the next round.
However, Butcher remembers how difficult their task was, not least because his side had to play the last 20 minutes a man down after Paul Cairney picked up a second yellow card.
The former Motherwell boss said: "They have been unlucky with a few things that have gone against them but they still have a good squad, still got some good players, players who are capable of hurting you.
"We know how hard we had to work to beat them in the cup and we don't want to go down to 10 men again like we did.
"We certainly want to play better and have that same steel and resilience that we had when we won in the cup."
The game, however, will take place under the pall of the tragic death of Easter Road youth player David Paul on Sunday.
The 18-year-old passed away at his home in the early hours before 13-year-old Jamie Skinner, who had left the Hearts Academy a couple of weeks ago, died after collapsing on the pitch playing for Tynecastle Boys club under-14 side.
While Edinburgh football continues to mourn the tragedy Hibs skipper Liam Craig admits he is "embarrassed" to even speak about football
Craig, who played for Tynecastle Boys club, said: "At times like this football doesn't matter.
"I am a bit embarrassed to talk about football when two families have lost their sons.
"As Hibs captain and a Hibs player, you are never prepared for this.
"I heard the news on Sunday and sat in a daze for most of the day, and then the news came though about young Jamie as well and Tynecastle is a club I played for at under-13s.
"The important thing is that we remember David and the club will do that.
"It is important that we are there for the younger players that played with him day in, day out.
"But the reality is we have a game of football on Boxing Day and we will do everything we can to win that game.
"We need to stick together and be professional and we will be ready for it.
"We are going up there to win the game and afterwards help each other through it."