Snails can suffer from a number of health problems and, unfortunately, there are not many veterinarians (if any) that will treat snails. Prevention is always better than, so it important to be aware of any potential problems so you can prevent them before they occur.
Below are some of the main problems you might come across, and I've included links to give you more information, but PetSnails.co.uk has very in-depth descriptions of many more snail health issues, as well as how to treat/prevent them.
Aestivation is when a snail creates a membrane and seals itself into its shell because of the dryness of its surrounding. This can be combated by increasing the humidity in your tank - easily done with a daily spritz of water from a spray bottle. Humidity in the tank should be around 75-90% - find out more about humidity monitoring here.
If your snail is still sealed in after increasing the tank humidity, you can encourage it come out by bathing it in lukewarm water, or by spraying the snail directly (but gently) with the spray bottle. Snails can take a little while to adapt to the new humidity level, so don't worry if they hide again soon afterwards.
Snails may also seal themselves into their shells if the temperature of its surroundings is too low. Most African snails require a temperature of around 20 - 29°c, although difference species can prefer lower temperatures. PetSnails.co.uk has detailed care guides for many different species of snail here.
A heat mat is useful for keeping the temperature of your snail habitat steady and warm. You can find out more about heating and heat mats here.
Snails can sometimes retract themselves into their shells and be reluctant to come out again. If there is no covering across the shell opening, then this is classed as retraction rather than hibernation or aestivation.
There are many possible reasons for retraction, and there is detailed information on these and how to treat them here.
Cracked or Broken Shells
Snails grow their shells by absorbing calcium through cuttlefish bones and food. Poor calcium intake can result in weak and flaky shells, as well as make them prone to cracking and breaking. Breaks on the wider part of the shell (next to the entrance where the snail emerges) are not too serious and the snail will usually repair these themselves in time. More serious breakages can occur on the point of the shell, where the older growth is. The snail cannot repair these, and if the snail's body is exposed, this can lead to other health problems, and drying out.
Shells can be repaired, but care must be taken to avoid the snail's body coming into contact with any materials or chemicals used. A guide to repairing shell breakages can be found here.