My guess is nixoncare would resemble in many ways the swiss/dutch healthcare programs with their focus on using insurers. At any rate we'd spend 40% less than OTL on healthcare, plus have a less topheavy/monopolistic business environment without OTL's system that de facto only allows SMALL companies or megacorporations to exist, so more medium-sized companies.
Effects on public support/policy? Well, you'd have the small government types drastically weakened in the GOP, plus a faster shift of working class whites to the GOP with effects on policy. A GOP that's got a bigger working class component, combined with a populist wing. The middle class yuppie social cons/movement cons would at most be a wing of the party with a not clean, but at least not caught Nixon being able to seal the deal with making a populist wing/moving the south GOP early, combined with no destruction of the moderate wing post-watergate, so a significantly less socially conservative/prudish GOP than OTL.