Improving or keeping their health up is an interesting challenge at times. I've tried a variety of things, and I'm not positive what's actually working and what's my imagination ;-). But I'll share a few things while also trying not to give too much away to spoil anything for anyone.
In the initial part of the game, trying to keep the food number as high as possible seems to help, combined with continuing to improve the healing and research. I also change their activities from time to time. Once you've reached Level 3 with Harvesting, it's much easier to keep the food numbers up, but it still needs watching so the numbers don't diminish too far -- I try to keep them over 1,000 at that point, but for Level 1 and 2 I aim for at least 300 at all times. Having no food does seem to impact villagers' health if the food isn't replenished quickly.
I've been able to improve health by being diligent about ALL these things, even going from having the red bar to being green all the way across. I also check on my villagers' health often and make adjustments in activities. If someone's green health bar slips a little I do some extra tweaking. It seems like balance is a real key to keeping them healthy -- food, healing, research, etc.
I'm still very much a beginner at this game, so the above is just my 2 cents at this point. I hope that helps a bit.