It won't happen in reality. In the real world, companies don't tell you how much they pay towards your insurance premiums, and all you see is what you pay out of your check towards it. Companies are not going to magically add it into your paycheck. With Obamacare, they will just claim that they are paying it in taxes to the government for your healthcare. I guarantee that no large company is going to give any of the premiums that they pay out to their employees. It won't happen, as it is an elective, not mandatory benefit.
Single payer IS government managed. You cannot have a single payer private solution that is not - that would be a monopoly, and as you should be already aware, those never work in favor of the consumer/citizens if they are not government managed.
Companies offer health benefits for two main reasons - to attract employees and to keep employees. It's that simple. It is in the employers best interest to attract the best employees and to retain them. It's the same as paid vacation/sick days, life/dental/medical insurance, pension, 401K's, etc - these are benefits offered by the employer as part of a compensation package. Traditionally, they have not been required to offer benefits like health insurance.
Why get rid of them? That should be up to the private employers who are giving them as compensation packages to attract employees that they deem valuable to their business. Regulation of compensation provided by employers is a start down a dangerous path. What's next, government controlled and mandated salary caps? How is it the government's role to determine what a private company can choose to pay an employee?