Kompany was taken to hospital after collapsing as he watched the club's second team train.
He had been unwell with a virus and missed the pre-match news conference on Tuesday for Bayern's Champions League game against Arsenal.
Kompany was on the touchline for Wednesday's match and Eberl questioned whether he "should have put the brakes on him earlier".
"The fact is, we protect our coach wherever we can," he said.
"If a coach says he feels ready to do something, then we're not going to say 'we won't allow that' if we don't know what's going on.
"It's also everyone's own responsibility to say, 'I can't carry on now'.
"Vinnie's a hard worker. If you don't listen to the subtle signals your body sends, eventually it will scream out in pain.
"Vinnie will be given that time [now], and we'll continue to try, even more than before, to protect him from himself."
Kompany spent 11 years at Manchester City as a player, winning four Premier League titles, two FA Cups and four League Cups, before going into management with Burnley.
After guiding them to the 2022-23 Championship title, Kompany was unable to keep the Clarets in the Premier League.
He left to take over at Bayern in May 2024 and has led them to successive Bundesliga titles.
Kompany's assistant Aaron Danks has spoken with the Bayern boss in hospital.
"I explained to him that we would be having no football conversations," said Danks. "It was just checking in on his health and making sure he's OK.
"All the football operations are in our hands and we'll work together as a team to keep the club moving forward."