Left hand squeezing is a common dilemma for student guitarists. It happens naturally: we try really hard to play the music set before us. It is replete with barre chords, difficult reaches, simultaneously held and moving voices. We try to avoid the ubiquitous fret buzz. Further, many of us came to classical guitar from playing other styles, notably popular styles on steel string or electric guitar. These styles often rely on blocked left hand shapes, learned as units, not as individual notes. The net result of all of it is often pressing too hard on the strings, and for too long.
Why does this matter? A couple of reasons come to mind. First, is that the harder you press, the more difficult it is to release, and therefore, the slower you will play. Think of this analogy: I was rear-ended not long ago by a tailgating driver. I wasn't hurt but when the jolt of the impact quieted, I found my hands were gripping the steering wheel so hard I couldn't easily let go. I could have used some help peeling them off the wheel! In the same way, when we over-grip the fretboard, our much-stronger flexor muscles (the ones doing the gripping) aren't easily overcome by our weaker extensors and so we tend to feel "stuck."
Next, there is an inherent fluidity in virtuosic playing that very specifically avoids pressing too hard, ever. The resultant looseness of technique coincides nicely with the quickness necessary for music both fast and complex, and feels and looks elegant, effortless, and balletic.
So how does one loosen their grip? I will approach the problem from two angles. The first is the purely physical.
Play a scale (no open strings) or a quasi-chromatic scale (four adjacent frets/string, across the fretboard). With the right hand, use i-m in alternation, quite loudly. With your left hand, press so lightly that you create a "thud" effect, mostly devoid of pitch. I call this a "thud scale." Run the scale up and back in this fashion.
Then press ever so slightly harder, to create a "buzz" effect. This is not the buzz that results from mis-fretting, too far from the fretwire, but is the result solely of pressing lightly. To really demonstrate mastery of this buzz technique, you must induce the buzz to last. It is relatively easy, if playing loud and pressing gently, to get the note's initial impact to buzz. It takes a bit more control to make the buzz ring on, for the full duration of the note. Doing so requires that you minutely adjust your pressure lighter and lighter as the note rings. Pressing too hard at any moment will stop the buzzing effect.
Using this long-ringing buzz technique, play slow scales up and down the fretboard. Remember to always play loudly with your right hand.
Play thud scales then buzz scales each day. It doesn't need to take very long. You are training both a divergent degree of pressure with left and right hands simultaneously and a minutely controlled and highly specific amount of pressure in your left hand. Take a moment and determine how much more pressure it takes to make the (loud) buzzing note sound normal. Alternate, a/b style, buzz then normal notes. You will develop a better sense of exactly how hard you have to press to get a good, clean, loud note. Obviously, if the note is softer, you can use less pressure. Working this method will help you to develop a light and highly controlled left hand touch, even when playing firmly with the right hand.
The other angle from which the over-pressing question needs to be addressed is more contextual and passage-specific, but can be addressed here generally: It's important to understand what notes you're playing need to ring, and thus be held, and which ones don't, and thus pressure on them can be released. This can seem to be an infinitely complex challenge, if minutely analyzed, and the motor skills necessary to control the left hand in this way are considerable. Fortunately, though, for most guitarists, these adjustments are made more or less automatically as the hand senses when a note can be released. We get into trouble when the textures are more dense, as in a Bach fugue. I have a student who struggles with this issue in the PFA Fugue. He is concerned that voices may not be given their full value and so overcompensates by pressing every note down as long as possible. Doing so makes shifts to new locations or positions difficult though, and so the piece ends up more challenging to navigate and sounds more clumsy and halting. We work carefully on identifying at exactly what moment certain fingers can be released, effectively weight-shifting to the remaining one(s) and enabling a more balletic feeling in the hand and a more legato effect in the interpretation. I see this quite a bit: a difficult shift or passage which resists a legato reading is corrected simply by realizing that a specific finger was holding too long. Look for these instances: it may make all the difference.
And by all means, lighten up!