Placed the finger on the fret? Should be behind the fretwire. And, there really should be no need to squeeze the heck out of the neck.
Yes, turning the index finger so the bony part presses down will help, but consider that it rarely, if ever, needs to press all 6 strings. Other fingers will be slightly closer to the bridge, often covering 3 of those strings.
If you're pressing hard with your thumb, that will maybe work, but it's not the cleverest ploy. In fact, you ought to be able to play a barre chord with no thumb touching the back of the neck. By trapping the guitar body with the right arm, and pulling in with the left hand.
The problem may be an action that just isn't that good. And trying to play barre chords on the 1st few frets isn't the way forward. Get used to barring from 5th fret upwards, until you're happier with the sound.