As a non-native English speaker I'm not too familiar with the expression 'pushing values', but I think the intent goes both ways, that is to say, it doesn't mean pushing them in one direction, but in both (or all) directions.
Pushing one's values too far either results in a high-contrast image, where all values have been replaced by binary values (resulting in a kind of poster decal art style), or is simply not really possible (within the use of this expression as a technical criticism), since values can only be stretched to a certain limit and can therefor not go too far.
It seems to me this is a matter of semantics, more than it is of art theory.
'Squeezing one's values', or rather 'limiting one's values', has a somewhat opposite meaning, and is illustrated well by the left part of the painting in the picture of the accepted answer in the question you linked (here).
Limiting the tonal range makes the image look hazy indeed, as the contrast has been limited, and this is exactly what happens when looking at objects through fog, or mist, or smoke, or from a larger distance (see also atmospheric/aerial perspective), or in the twilight, where the additional effect of diminishing colours can be appreciated even more clearly.
This expression ('squeezing values') deviates slightly from being an exact opposite of the former expression ('pushing values'), in the sense that these values can be 'squeezed' or limited within several value ranges (it is not a binary adaptation), e.g. the edited images on the left and right of the centered original below:
Euan Uglow | Two Apples (source). Here the Curves tool in Photoshop was used to limit the colour range
As you can see, the difference in atmosphere in both edited images is dramatic: the first seems to want to convey a certain gloominess; the second gives the impression the artist wanted to show the intense and warm lighting of a summer's day. (Whereas the original exemplifies Uglow's generally quite neutral palette - although any image naturally would have its most neutral-looking version in between these renditions.)
Limiting your values like that will usually let lighting play a more important role, but everything will depend on the (intensity of the) colours one chooses: colours can become desaturated when changing their value, but you can also opt to only work with colours within a certain value range to best maintain their saturation.
To get an idea of what can be achieved with limited value ranges, I suggest looking at the works of Giorgio Morandi specifically, and several symbolist painters (like Pierre Puvis de Chavannes), some (renaissance) frescoes (esp. Pierro della Fransesca), and maybe Vilhelm Hammershoi for his twilit atmosphere.
Unfortunately, I can't yet think of a good example of an artist working with predominantly dark value ranges.