So rather than having the girls become magical girls, in this universe the girls can be recruited to the Lantern Corps.
Homura (Violet): Homura was chosen by the Violet ring because of her love for Madoka. However, her love for Madoka became an obsession that drove her insane thanks to the ring’s power. She tries her very hardest to prevent Madoka from joining any of the Lanterns, so as to prevent Madoka from being killed by the War of Lights or Blackest Night.
Mami (Green): The Green Lantern Corps chose Mami to join their ranks after they discovered her desire to do good. She discovers Madoka and Sayaka as civilian and decided to have them join her on her many adventures. She did bump heads with Homura at first, but did not realize it was because of Madoka. She was killed during a skirmish with the Sinestro Corps, and Homura stepped in to keep Madoka and Sayaka safe.
Sayaka (Green —> Black): When Mami died, her ring called upon Sayaka to be the next Green Lantern. Like Mami, she to waned to fight for justice an was revered as a great Green Lantern. However, she fell into despair during the Blackest Night and became a Black Lantern. She was destroyed by Kyoko.
Kyoko (Violet): She was chosen to be part of the Star Sapphires after showing great love and devotion to her father’s cause. At first, Sayaka used the power of her ring to recruit more members to her father’s church, but was called out on it when her father found it. Her father feared that Kyoko turned into a demon and abondoned her. After some time, she discovered Sayaka and fought with her multiple occassions, since Violet and Green Lanterns had some squabbles. Before the Blackest Night event, however, she and Sayaka got along after talking about their lost loves. When she discovered Sayaka turned into a Black Lantern, she decided to give her life to destroy herself and Sayaka.
Madoka (White): Madoka was chosen to be a white lantern to put an end to the Blackest Night.