Are employers responsible for offering health insurance to employees through companys of the employees choice?

Employers are not responsible for offering or providing health insurance. If they offer and provide health insurance, it is simply a perk (which is taxable in the UK). You are limited to whatever the policy and company they provide and is usually only provided to the employee. If you wish to hold a different policy with the same health insurance provider, you are typically expected to pay the difference. Adding members of your family to the policy will also typically not be covered by your employer so you will have to pay for them.