I wasn't able to figure out the issue. It appears to be a really poorly done image. The deprecated Docker image seems to actually have eagercli unlike this Singularity image.
If you are using the newest Singularity you can try the steps below to pull the Docker Hub image and convert it to Singularity. The image will be named eager.img by default.
singularity pull docker://apeltzer/eager
If you then execute singularity exec eager.img eagercli you should get an actual EAGER error.
Note: They do say the Docker one is no longer supported.
We had some issues with EAGER in Singularity that had a missing eagercli - these have been fixed for a while. Opening a GitHub Issue would have helped me to spot that earlier on, since we're using a dedicated installation at our facilities without Singularity as a container solution.
You should be able to download via Singularity without the docker intermediate step and things should work. Again: Always ask authors on GitHub via a Issue ticket - not everyone has a Biostars account, checks back here frequently and can help you out.