When private companies offer health insurance to their employees the insurance carrier they contract with typically requires that all employees be obligated to participate and pay premiums whether they wish to or not

What problem is the health insurance company trying to avoid? How does this policy mitigate this problem?

The problem that the health insurance companies are trying to avoid is that of adverse selection. Those who believe they are likely to need health insurance are those most inclined to sign up for benefits. Those who are less likely to become sick or perceive themselves at risk would be less enthusiastic. If there were no obligatory participation then health insurance companies would likely be insuring only those workers most likely to make claims. This is not a very profitable way to run a health insurance company.