The 40-year-old actor recalled the moment during Thursday's episode of Good Morning America after sharing a video of himself visibly nervous to greet the royal couple.
"You can't see, but I'm standing next to Seth Rogen," Eichner said. "Before you meet them, they e-mail you this whole list of very official protocol of what you're supposed to say, and your royal highness and don't speak until you're spoken to, but we didn't really know if they wanted us to do that. It feels a little strange and we're American, so I was panicking about what to say."
"I was turning to Seth, I was like, 'What are you going to say? What are you going to say? What are you going to say?' I think just muttered, 'Oh, pleasure to meet you,' and shook hands. They were lovely," he added. "And Meghan and I went to Northwestern [University]. We went to the same school. So we just ended up having this very casual conversation about our acting teachers."
Eichner had posted the video of himself Monday following the premiere.
"OMG Here I am completely freaking out to @Sethrogen about how I'm going to greet Prince Harry and Meghan I'M CRYING," the star wrote, adding several tears of laughter emojis.
"BTW they were both super down to earth and lovely and Meghan and I talked about our @NorthwesternU acting teachers (shout out to Mary Poole and David Downs!)," he added.
Eichner and Rogen voice Timon and Pumbaa in The Lion King, a Disney remake of its 1994 animated film of the same name. The movie co-stars Donald Glover as Simba, Beyoncé as Nala and Chiwetel Ejiofor as Scar.
The Lion King opens in theaters Friday. Beyoncé will release the album The Lion King: The Gift the same day, which she described this week as a "love letter to Africa."