Many big companies self insure. They pay the insurance company to manage the claims, and to have access to their network of doctors, hospitals, specialists, and pharmacies; but cover the costs on a shared basis with the employees. Medium sized companies use one of the standard group policies. Small companies either have expensive policies because they are a small group, or they have to join with other small companies through an association to create a larger group.
The bigger the group the less impact each individual person has on the group cost. The insurance companies reprice their policies each year based on the expected demographics of the groups, the negotiated rates with the network of providers, the required level of coverage, and the actual usage of the group from the previo year.. If the insurance company does a poor job of estimating the performance of the group, it hits their profits; which will cause them to raise their rates the next year which can impact the number of companies that use them.
Some provisions of the new health care laws in the US govern portability of insurance regarding preexisting conditions, minimum coverage levels, and the elimination of many lifetime cap. Prior to these changes the switching of employers while very sick could have a devastating impact on the finances of the family. The lifetime cap could make it hard to cover the person if they had very expensive illnesses.
If the illness doesn't impact your ability to work, there is no need to discuss it during the interview process. It won't need to be discussed except while coordinating care during the transition. There is one big issue though. If the old company uses Aetna, and the new company doesn't then you might have to switch doctors, or hospitals; or go out-of-network at a potentially even bigger cost to you.