Assuming that you are talking about Squirrelflight's big lie, Squirrelflight lied to protect Leafpool as well as Jayfeather, Lionblaze, and Hollyleaf from the truth.
Squirrelflight pretended to be their mother because, she was protecting her sisters reputation, life, and huge secret. Brambleclaw didn't know the kits weren't his, so he didn't lie.
Squirrelflight lied to help her sister because Leafpool was a medicine and accidently had kits and was wrong to have them so to cover it up she asked her sister to pretend they were hers so her clanmates along with the rest of the clans wouldn't know.
Squirrelflight is Leafpool's sister!
Squirrelflight's father was Firestar.
Ashfur did not kill Squirrelflight in any book. He did attempt to harm her in "Sunset" (Warriors: The New Prophecy series) by attempting to push her off a cliff, but Squirrelflight survived. Ashfur's actions were driven by his jealousy and betrayal, but they did not lead to Squirrelflight's death.
no squirrelflight WAS mates with Brambleclaw but not anymore,and ashfoot is a windclan she-cat
Squirrelflight is not infertile. She and Bramblestar have kits together in the Warrior Cats series.
Nobody really knows except the Erins and they will never tell until they write it in the books. So far Squirrelflight seems to still have affection for Brambleclaw but he does not show the affection back. Their relationship is becoming stronger but they would sooner be friends then mates by how they act together.
they didn't squirrelflight lied.
No, Leafpool is not the mother of Squirrelflight's kits. Squirrelflight's kits, Alderheart and Sparkpelt, were actually born to Squirrelflight and Bramblestar. Leafpool is Squirrelflight's half-sister.
Firestar and Squirrelflight
Yes, Squirrelflight and Brambleclaw knew that Jayfeather was not their biological kit. Squirrelflight had taken Jaykit (Jayfeather) from Leafpool, her sister, to raise as their own after Leafpool was unable to care for him due to her role as a medicine cat. Brambleclaw was also aware of the situation and supported Squirrelflight in this decision.