The track, titled "Ta liberté de voler" (Your freedom to steal) takes aim at sites where users can download music and video for free, reports Le Parisien newspaper.
The group's manager told the newspaper the band was "expressing things that other artists don't dare say."
Megaupload was shut down by the US Justice Department in January following the indictment and arrest of its owners for alleged copyright infringement.
Referring to Kim Dotcom, the German founder of Megaupload, lead singer Ivan Callot said he found it "incredible" that people were ready to give money to a "well-known crook."
The song has sparked a strong reaction online since its release on the website dailymotion.fr on Tuesday.
Callot said the song was "not attacking people who download, but those who wrongly champion such sites thinking they're helping culture by making it freely available."
"You're a bit of Robin Hood, Mesrine and Jesus all at the same time," go the ironic lyrics of the song. Mesrine refers to an infamous French criminal who died in 1979. "You clicked and became a revolutionary."
"Thanks to you, the world of music has changed. Small artists at last appear on TV" the song continues, before "oh no, it's the reverse that's happened."
Les Rois de la Suède was formed in 2009 by Ivan Callot, a singer who founded another group, les Fatals Picards, in the 1990s.
The group claim to be "musical ambassadors of the Swedish savoir-vivre."