Oil the inside flaps, see if that helps. They really wax up leather to go into the show room, so rather than just oiling it you might want to try a 'system' product.... I.e. Comes in 2 bottles: a cleaner, and then a conditioner. I forget the brand of product I use, but that works the best. You clean it and then use the conditioner sparingly until no residue is coming to the top (for new leather it usually takes cleaning it twice and then conditioning 2-3 times with just a few drops of the stuff (goes a long way)...)
But I would first try cleaning the inside flaps (under the saddle flap, the billets, and the top of the sweat flap).
Another thought (though unlikely), is it creaking rather than squeaking? Something in the tree could be broken - but it's a different sound than a 'squeak' like too-clean, waxed up leather.
Synthetic saddles are PRONE to squeaking too. I swear the get a build up of something on them.