Why Isn't Big Business On Board with Single-Payer Healthcare?

This is a question that I ponder when I read about employer complaints about the rising cost of healthcare. This article on work being done by Labor and the Auto industry to change national healthcare policy (noting that $1500 of ever car and truck GM makes goes to healthcare costs) is interesting, in that it proposes national measures.

But when management complains about their competitors' nationally subsidized healthcare costs (Toyota only spends $186 per vehicle on healthcare), why don't they make the (short) leap and support national healthcare in this country? Is it because they have an ideological problem with government subsidies (except for energy)? Is it because they are somehow secretly beholden to the healthcare industry (even I cannot concoct that conspiracy, but I welcome a reader to try)? I really don't get it.