Dems keep saying "if you like your insurance you can keep it" but...?

To sell the universal health control, Democrat politicians are telling us we can keep our private insurance - making us think we have a choice. But isn't it likely that if it's free, the majority of freeloaders will choose the gov't controlled side, causing private insurance to fade away ? Is it fair for gov't to compete against private companies since they have the force of law and can confie our income?