Killie are looking to move six points above the Scottish Premier League's bottom club when they meet at Rugby Park.
A Kevin Kyle hat-trick earned the Ayrshire side a comfortable 3-0 win over Falkirk last month but the Bairns have since cut the gap.
Jefferies, who managed Hughes at Falkirk, believes criticism of his former player from a section of fans is unfair, one season after the club were eight minutes away from making the top six.
"John Hughes has done a fantastic job there," Jefferies said. "Nobody knows him better than Billy and I as a character and a winner and how he hurts and how he'll fire his team up.
"He'll be disappointed with the last performance but they have rallied from some of the criticism.
"They got a draw with Hearts, then he goes into a national final and got a win against Motherwell last week.
"They have responded in the right way. We have to make sure we get out there and focus on our own game and we know we have the advantage that when we do play well and do compete as well as we can, we've shown Falkirk just how good we are.
"But he has done a terrific job there and all the criticism he has been getting is just down to expectations."
Falkirk needed a point at Aberdeen to clinch a top-six place last season but lost a late goal in a 2-1 defeat.
"Because they just missed out last year, they (the fans) expect them to be in the top half and not struggling like they are," Jefferies said.
"The praise comes along but we all suffer from criticism. It's part and parcel of the game; if you don't like it, you shouldn't be in the game.
"You just have to be thick-skinned and stick to what you believe in and trust your players.
"I think I've got enough in my squad to know we can beat anybody on their day. Hopefully, we can do that enough times in the last four games to get safe."
Killie host Inverness and St Mirren following the visit of Falkirk and Jefferies has urged his men to capitalise on home advantage.
"We've said you'll not get a better chance with three games at home," Jefferies said.
"We won our last home game so we got the monkey off our back a wee bit there. The fans will get right behind us for these three games.
"If we beat Falkirk, we go into the next two home games back-to-back with six points with three games to go.
"That's a great advantage with the home games and the lift we'll get if we beat Falkirk."