No, health-insurance mandates will not help solve the problem of free riders, because the mandates can only be enforced if someone receives a tax return. Even so, a person can just pay the penalty until they have something go wrong, and then they can apply for insurance, because pre-existing conditions cannot be denied.
Health insurance mandates do not solve any problems. There will always be free-riders and mandates are not going to stop that. There will always be those ignoring the mandates and paying the fees. There will always be people who know how to work the system. Mandates do not do anything except make the law abiding citizens feel there is one more thing they have to do.