Health insurance and healthcare is NOT tied to your employer in the US.
However, many employers pay 50% to 70% or even 100% of the health insurance premiums for their employees, and employees can use pre tax income to pay health insurance premiums if purchased by their employer.
The problem has always been this tax advantage of your employer purchasing it, AND most Americans opting to not purchase it due to cost.
Note that there are significant subsidies available if you earn less than 400% of the federal poverty level, although that may still leave many unable to purchase it (or the deductible/oop max is so high it would not make any difference).