No, businesses should not have to directly pay for health care for several reasons.
First, forcing businesses to pay for health care for all employees greatly increases the cost of each worker to the business. Even basic health coverage adds thousands in premiums for each worker. This money has to come from somewhere. Businesses will respond by either reducing pay (when possible with minimum wage laws), raising prices to offset the increased labor costs, reducing staff and requiring more productivity from each worker, or moving operations to places with lower labor costs, such as Mexico, China, and India. All businesses, regardless of industry or size, would implement some or all of these measures immediately if required to pay for health insurance on each employee.
Second, different businesses would opt for different plans and coverage levels, so changing jobs could impact just how much medical care a family can afford. Of course, this change is still better than the current prospect of losing insurance completely, but it is still undesirable to suddenly be unable to pay for your prescription or other treatment because of a difference in coverage.
Third, shifting costs to employers shields workers from truly understanding the costs of their health care and can encourage frivolous spending. It is much more tempting to indulge your inner hypochondriac when each visit to the doctor is only a $15 co-pay and your insurance hides the rest of the cost. This unnecessary spending would only continue to push medical costs through the roof, with no real benefit to overall health.
So overall, forcing businesses to pay for insurance for all workers will make American businesses less competitive, put downward pressure on wages, reduce job growth, and increase overall health care spending. At the same time it would fail to eliminate the financial strain of changing insurance coverage from job to job and encourage wasteful spending.
A much better option would be to have a system where consumers shopped for their own coverage, like they do for life, auto, and property insurance. In a large and competitive market, consumers could select the coverage that suited them best from a variety of offerings by different insurers. Tax breaks and employer contributions, a la 401(k) could mitigate the out of pocket expense. In this scenario, workers would be able to pick plans that gave them the coverage they needed, but with consideration to the overall cost. By separating employment from insurance, unemployment or changes in employment no longer means losing or changing coverage.
For me, the choice between forcing costs onto businesses during a time when global competition is fiercer than ever, and giving individuals the freedom to select the best option for their unique situation is really no choice at all.