I live in Utah, but I just got off the phone with my brother who lives in Nebraska.
He has been working for a company as a supervisor for the last year, and was just fired. According to him, he was laid off because someone in HR found out that he had a felony conviction (I believe it was approximately 7 to 8 years ago). He fully disclosed the facts regarding his conviction both on his application and discussed it in his interview. Again, according to him, there is no other reason why he was fired. Is this legal?

In that no law prohibits an employer in your state from firing an employee for a conviction record, yes, it is legal.

However, unless there is an upcoming contract or something of the sort which would require all employees to have clean records, what you describe sounds very unlikely as he has described it to you.

I can almost offer you a guarantee that there is something he is not telling you.