As far as I know there are no health risks to using CRT over LCD. Both produce electromagnetic radiation, but not enough to harm you, even through longterm use.
As long as you don't use the windows default refresh rate of 60hz, you will not get any more eye strain using a CRT over a LCD. The usual cause of eyestrain isn't caused by your screen, but by the way you use it. Some people have the tendancy to stay focused on the screen and only the screen rarely looking away. This definately causes eyestrain. periodically refocusing to places other than your screen reduces the probability of eyestrain.
So, if you're looking to justify buying a new LCD over your CRT, eyestrain really isn't a valid reason and you'll need to look elsewhere. There are plenty of compelling reasons to go LCD over CRT. Power usage, Deskspace, Widescreen displays.
IMHO the only real advantage CRT has over LCD is that it displays color better, especially blacks and is still cheaper than LCD.