The Toffees have experienced a miserable start to the season so far, as they are winless in the Premier League as well.
While they managed a draw against Southampton in normal time on Tuesday, they lost on penalties. Veteran Ashley Young had his spotkick saved for Saints to win the tie.
"That is the tough thing about football, it is a tough industry," Dyche said of the reaction to the game by the fans.
"The crowds have opinions about varying players and that is one of them things. He is certainly a tough customer mentally.
"I totally understand it, Young is old and wise enough to know this is the way the industry is. He is a fantastic professional but the fans have a right to say what they want to say and I am never going to question that.
"Inevitably we wanted to win the game and we haven't done. After piecing together a team this morning, with three players going down ill overnight, I think we have given as good as we can get.
"We have got 12 outfield players currently fit."