Yui only appeared during a sidestory in the second light novel so we can say that she didn't play a role in their decision.
Basically Heathcliff knew that Kirito and Asuna felt a responsibility for clearing the game seeing how they were some of the strongest players.
Asuna then suddenly removed her smile and muttered:
“Up until now, I had only fought on the upper floors. I had
completely forgotten that there were people who lived normal lives…”
“I don’t mean that we’re special; but since we’re on a high enough
level to fight on the front lines, I suppose it also means that we have an
obligation to them.”
“…I had never thought about it like that… I always felt that
becoming stronger was just a way to survive.”
“I think that a lot of people are going to be counting on you from
now on. Of course, that includes me.”
“…except with my personality, hearing these kind of expectations
only makes me want to run away.”
As Asuna pouted in dissatisfaction, I stroked her hair and hoped
for this life to continue for a while longer. For Nishida and the other
players, we have to return to the front lines at some point. But at least
Based on the messages that Agil and Klein had sent me, I knew
that they were having a hard time trying to clear the seventy-fifth floor.
He did guilt-trip them a bit with his message:
That night, we received a message from Heathcliff, requesting us
to take part in fighting the seventy-fifth floor boss.
Seeing as we had temporarily left the guild, we could decline the
request. But the last line of the message, that “some people had already
died,” laid heavily on our minds.
Both excerpts are from the first light novel Aincrad (translations by Baka-Tsuki).