Did they cite the study(s)? I think your nurse maybe had the newish Barouch paper in mind, which if I understand right looked at older bivalent vaccines and only showed increased longevity for COVID response, not flu. There's many other papers that show no change in longevity for coadministration.

It's great to get vaccinated for both and I applaud the thrust of that message, but some people may need to time things differently depending on how their bodies react to these vaccines, and I don't know if there's enough evidence yet to firmly recommend coadmin.