In both cases the eyes are in prolonged exposure to relatively very high energy spikes in the spectra emitted by black body ultra-violet light sources such as energy saving White LED and CCFL.
1) LCD users are perhaps slightly better protected due to there being various filters in the displays but these filters are not perfect and are of varying quality, meaning that some displays give you eye strain quicker than others. Specs do not detail emitted spectra (of eg. a "white pixel" at certain standardized calibration) so this is very hard to factor into buying decisions.
2) Prematurely born babies with under-developed eyes lacking protection and staring directly at bright overhead lighting in hospitals having not even developed reflexes to close the eye lids offers somewhat analogous setting to staring at LCD CCFL's. It's worth noting that eye lids offer no protection - this is why I feel strain sensation even eyes closed when sitting in front of LCDs - only putting my hand on top of the eyes will give relief.
Spikes in various datasheets of CCFL and White-LEDs may look insignificant, but this may be result of averaging eg. 10 nm band instead of reporting the peak intensity by having a technique to find/focus to the peak of the spike.
Of particular concern is evidence that the human eye is most vulnerable to high intensity light at blue light frequencies, where energy spikes from CCFL and White-LEDs occur.
I don't know how exactly the eye controls how much light it takes in but I have reason to believe that it does not adapt to the maxima of these spikes. If it did adapt to the maxima, I reason that that would cause you to only see the spike and everything else would look dim. Instead I believe it adapts to the broader average spectra intensity and lets these spikes in at full strength and that's my current best theory on why I get eye strain from CCFL and White-LED but not so much from CRT's and not at all from incandescent light.
CRT phosphor spectra, much more balanced and not a spiky: - I can watch CRT nearly full day and only start to get eye strain toward the end of the day. On worst LCD (Samsung phone) I get epic eye strain within minutes!
(CRT = dashed lines, the red phosphor appears to be spiky but this is not as big problem since according previous graph, eyes are much better protected in these wavelengths)