Eta takes charge after the dismissal of Steffen Baumgart, who was axed after Saturday's 3-1 defeat to bottom side FC Heidenheim.
Eta will be caretaker manager to the end of the season.
The 34 year-old first joined the senior staff in 2023 and has been on the touchline during a three-match suspension for former coach Nenad Bjelica in January 2024.
Eta has been named the club's womens coach from the end of this season. In the meantime, she will be in charge of Union's mens team to the end of this term.
"We have had a hugely disappointing second half of the season and will not allow ourselves to be blinded by our league position," said Horst Heldt, Union's director of football.
"Our situation remains precarious. The performances shown in recent weeks do not give us confidence that we can turn things around with the current set-up. We have therefore decided to make a fresh start."