Stephen Kenny’s side haven’t won one of the 12 Nations League games they have played since its inception, their latest defeat coming to Ukraine four days after defeat in Armenia.
Egan came off with a knock against Ukraine, which may keep him out of Saturday’s clash with Scotland in Dublin, before Ireland face Ukraine again next Tuesday in Poland.
And Egan said: "That's football. You get knocked back and it is all about how you respond to it and how you bounce back.
"We have a changing room in there where everybody wants to bounce back and put it right next Saturday.
"The players always give 100 per cent and that's the least you can expect when you put on an Ireland jersey. Everybody has given everything, we want to win the game but came out on the wrong end of a tight game.
"Listen, it is one game at a time. We have to try and get back into it on Saturday. Obviously it is not the start we wanted, but we have to keep going, keep looking at the next game."
The Irish started and finished brightly at the Aviva Stadium and Egan’s defensive partner Shane Duffy rattled the crossbar late on. But Ukraine’s second string – boss Oleksandr Petrakov made 10 changes to the side which lost 1-0 in Wales in Sunday's World Cup play-off final – proved too slick and quick-witted for their opponents.
Substitute Viktor Tsygankov scored the winner after Egan’s foul, from a free-kick which was allowed to bounce in front of wrong-footed keeper Caoimhin Kelleher of Liverpool.
Egan added: "It was a dangerous ball into the area between the defenders and the goalkeeper and obviously if you can't get a head to it, sometimes they fly straight in. It was a disappointing goal to concede.
"Overall I thought we played well enough and created a lot of good opportunities. We just couldn't take them."