How Can I Mute Squeaky Guitar Strings?

As the strings get older eventually they will 'squeak' less and you will get used to the sound. I remember when I just started playing and hated the squeaking sound of the strings, until I told my guitar teacher about it. He just told me that it's a natural part of playing acoustic and you can always hear it. Don't worry too much about it.
Besides I personally believe that it gives the guitar a certain sound which reminds me of a cello, which sounds very Minorish and beautiful.
In Incubus - Mexico you can hear how good it actually sounds with the squeaks of the guitar.

The strings will get less squeaky as they get older. Don't worry about it.

Not unless you use plastic or gut classical guitar strings (which are more expensive, but not by that much), and you can still hear some sliding around.  However, even most classical guitarists use metal wound strings on the three lower ones just for projection purposes.  You can cover the sound up on an electric guitar with distortion, but that's just masking it with another noise.  

Good technique eliminates a lot of the squeaking, but even the masters (Andres Segovia for instance) make/made squeaks whilst playing, if you use flat-wound or ground-wound strings, they won't squeak as much (but don't sound as bright as round wound IMO)



Segovia making some (small) squeaks ;-)  (A small one at approx 11 secs and another at 49 secs -I am sure you could find more if you listen carefully )

Engineering techniques to eliminate squeak on recordings would be to program an automation mute/volume cut on the offending track on your DAW, or to copy the offending part to a new track, and invert the phase of the offending squeak and thus eliminate it by phase cancellation.

The best advice I could give is to practice lots (and lots more ;-)  ), and to not get too frustrated at the noises that "escape by mistake"  ;-)

HTH