In a basic physics sense, the amount of force applied is the same regardless of how far down you go. A mass of, say, 100kg (just to pick a nice round number) under the influence of gravity (which provides a downward acceleration of 9.8m/s^2) generates a force of 980 Newtons (100kg * 9.8m/s^2 = 980kg-m/s^2 = 980N) downwards. If you have this weight on your back, you have to exert slightly more than that force upwards in order to lift it. This is the same whether you go 1/4 or ATG. What differs is the amount of work you do. In physics, "work" has a precise definition-- force applied across a distance, measured in joules (N-m). So if you squat in such a way that the bar moves 1 meter as you rise, then with the 100kg load in this example, you are doing 980N * 1m = 980 joules of work (ok, slightly more than that, since you have to beat gravity, not just match it, but lets ignore that). I guess it's obvious at this point that the amount of work done for a given weight is directly proportional to the distance moved. So if someone moves the bar only 1/4 the distance, they're doing only 1/4 the work, or in this case, 245 joules. So then if you want to figure out how much weight can be moved over the full 1 meter using only 245 joules, you basically divide it by the 9.8 number from earlier, which gives you 25kg. It's all neatly proportional-- moving an equivalent weight 1/4 the distance takes 1/4 the work, so doing that smaller amount of work over the full distance allows only 1/4 the weight.
However. A key point is that the 1/4 squatter using the same weight as a deep squatter is in fact exerting the same amount of force
on the weight as a deep squatter. But, this physics example is a simplication. It works for blocks being lifted perfectly straight. But the biomechanics of a squat are a lot more complicated. When you're down in the hole, your quads, parallel to the ground, have a lot less mechanical advantage then they do when you're partially upright. So if you squat deep, you actually do have to exert more force through your quads to transmit the same force to the weight. I think this is the crux of why deep squats are harder-- pushing out of a deep hole is mechanically disadvantageous, so you have to work harder to do it. But ultimately the force actually applied to the bar is the same, as per the first paragraph above.
So what does all this mean in terms of how much the 1/4 squatter could lift if he did a full, deep squat? Beats the hell outta me.

But I think it hinges more on the mechanical advantage penalty at the bottom then purely the distance traveled while squatting. The 1/4 squatter is capable, in the mostly upright position, of exerting the force needed to move the bar. This is exactly the same amount that the deep squatter is doing when passing through the same position. If it were possible to keep doing this over a longer distance without changing the position of the legs, the 1/4 squatter could probably do the necessary work (in the physics sense). But whether he has the "overcapacity", if you will, to get it out of the hole when the legs are sharply bent is the real crux of the question.