No idea whether this actually happened, but if it did, it couldn't have been 1977. Leadon left at the end of 1975. "Hotel California" was released in Dec. 1976. The Eagles third album was "On the Border" which was released in 1974. So if their third album was not released yet, that would have been late 1973 or early 1974. I'd have to check on the exact release date of OTB, but I know it was early 1974.
As to whether they were "too big" to be Neil Young's backing band, the comment states they played a full set and then brought out Young for the encore. That's not the same thing as just being his backing band. I have no clue now likely such a scenario is, but it was a benefit and not a regular Eagles show, so who knows what they might have done. The Eagles did something similar for an RAC benefit in 2003. The Dixie Chicks opened, then the Eagles played a shortened version of their regular show, then the Dixie Chicks came back out for the encore and they played several songs together.