Rimes's absence left the magazine scrambling, but January 2011 cover girl Brooke Burke instead filled in to host the party, which was held at Trousdale nightclub in Los Angeles, a source tells PEOPLE.
"Shape was excited to throw LeAnn the party," the source tells PEOPLE. "The people at the magazine wanted her to know they were happy with the cover and wanted to maintain a good relationship with her."
Rimes's absence caused a stir because of an e-mail sent out by Shape editor Valerie Latona to readers who were upset about the cover choice because of Rimes's controversial love life, in which she carried on an extra-marital affair with Eddie Cibrian. But Rimes's rep said the 28-year-old was too sick to show and that it had nothing to do with the affair drama or recent engagement rumors.
"Unfortunately [Rimes] caught the flu and under doctor's orders was advised not to travel or go out," her rep tells the New York Post.
Rimes herself tweeted Friday, "Headed to the Dr. in a bit. This flu thing is hard to shake. Just to clarify, I was never supposed to sing at the Shape party and yes, under Dr.'s orders I couldn't go. I spoke to Valerie yesterday I don't know how many times and all is well w/ absolutely NO controversy."
She added: "Just happen to get sick at the 'perfect' time, SUCKS!!"
The party of about 45 guests, regardless, had drinks from Ciroc vodka to enjoy. "Even though everyone kept cool on the surface when news of LeAnn's no-show occurred, the magazine wasn't sure how to approach the guests," says a source. "The editor-in-chief cracked a joke on a riser to the crowd about issues she and LeAnn have had in the media. But her comments seemed all in light-hearted fun."
Rimes has also postponed a N.Y. concert planned for Friday night at Turning Stone Resort & Casino Showroom.
With reporting by MARISA LAUDADIO