(Just in case of any confusion, this answer was originally in response to a duplicate question about pinch harmonics used by Billy Gibbons of ZZTop - thanks for the merge, Dr Mayhem!)
These are probably pinch harmonics. In which case, they are not produced by a particular amp or effects setting, but are produced by the picking hand while playing (although they are much easier to produce when using a lot of overdrive/distortion).
Pinch harmonics are produced by creating a node with the picking hand as it picks, rather than creating a node above one of the frets (eg. 12, 9, 7, 5) with the fretting hand, which are natural harmonics. (See the end of this post, for an explanation of what nodes are and how to find and create them.) There are several ways to do this. Most players do pinch harmonics using the thumb of the picking hand; I touch lightly with the nail of my ring finger (3rd) on my picking hand. Either way, the thumb or finger should touch at exactly the same time as the pick and then move away from the string at the same time as the pick. You will need to experiment with moving your pick position along the string, to find different node positions for different pitches (and, of course, these change, depending upon which fret your L.H. is fretting).
Like many players, I suspect, I first "discovered" pinch harmonics by accident (usually while using distortion). With a bit of practice, though, it becomes easy to create pinch harmonics for any fretted note. I must confess, I take a rather "non-scientific" approach to pinch harmonics, being able to find them somewhat intuitively. However, there are plenty of guitarists who know exactly where to create the R.H. node to create exactly the harmonics they want for any particular fretted note.
Another term for harmonics created when playing fretted notes is artificial harmonics. In particular, this term is used when playing classical guitar. Obviously, no pick is used for classical guitar, so the method of production is different. The node is created by the R.H. index finger; the string is then usually played with the R.H. ring finger. (Although, this may need to be different if playing chords containing an artificial harmonic.) When producing these kind of artificial harmonics on classical guitar a hit-and-miss approach is not useful; the R.H. index finger needs to be placed exactly at the node point on the string (usually half the length of the fretted string, to produce a harmonic an octave higher than the fretted note, although other harmonics can be effectively produced too). The lack of distortion and volume when playing classical guitar means that only "exact" harmonics sound effectively, whereas, with plenty of volume and distortion, pinch harmonics can almost be produced accidentally as a kind of "effect" when playing electric guitar.
ADDITIONAL INFO: What are nodes? How do I find and create them, to produce harmonics?
As requested, here is some additional info about finding node points on strings to create harmonics.
Node points for harmonics can be found at points along a string by dividing the total length by an integer (whole number) value. For instance, there is a node point at 1/2 way along the string (in other words, it is found by dividing the total length by 2). Dividing the total string length by 3 gives two node points, producing the same harmonic; one is a 1/3 of the string length from the nut, the other is a 1/3 of the string length from the bridge.
If a vibrating string is touched at a node point (while or after it is plucked) the string vibrates "in sections", with the node point (and other equal node points) being still. This creates a harmonic, a higher pitched overtone, present in the open or fretted pitch (the fundamental frequency).
The pitch of the harmonic is directly related to the integer value of the string division. For instance, playing the harmonic with a node at the halfway point along the string doubles the frequency, which makes it an octave higher. If the node point is found by dividing the string length by 3, the frequency is 3 times the fundamental frequency, which makes it an octave and a fifth higher. The wikipedia article here gives a good, detailed explanation (and illustration) of how strings vibrate when touched at these different node points, and how this affects the frequency of the harmonics produced.
How does this relate to playing pinch/artificial harmonics on guitar?
When playing an artificial or pinch harmonic, the node points are now a division of the fretted length of the string; in other words, the distance between the bridge and the fret being pressed with the fretting hand.
If using the R.H. thumb to play pinch harmonics, the point of the thumb closest to the string (usually the left side of the thumb's tip), touches the string at a node point, at the same time as the pick touches the string. The pick will be plucking the string slightly to the right of where the thumb touches the string. The images at this webpage illustrate this well.
If using a R.H. finger to play pinch harmonics, it's tip (or nail) will again touch at a harmonic node-point, but the pick will now be plucking to the left of where the finger touches the string. I use my R.H. ring finger to do this, but it is possible to use the index, middle or little fingers, too.
To produce artificial harmonics on classical guitar, the R.H. index finger usually touches the node point (by "pointing" somewhat towards the neck), while the R.H. ring finger plucks the string. The image below illustrates this well:
EDIT: one last point. The technique of using the ring finger to pluck the string is helpful on classical guitar, as it means you pluck the string some distance from where you touch the node. As classical guitar is acoustic (and so, obviously, doesn't use distortion either!) it is harder to produce a strong harmonic. Plucking further from the node helps to do this. I illustrate this in my answer to another question, here.