As a nitpick, the OP refers to “socialized medicine” which is a right wing pejorative for universal healthcare.
Universal healthcare does allow for people to work part time jobs and still have insurance.
/nitpick
What you might have is underemployment (regardless of how many jobs you have). And then, there’s still the matter of income disparity and at what level of income you have the poverty threshold.
a nitpick of the nitpick 
universal healthcare does what it says on the tin and is available to all regardless of job status.
One thing that would be affected are National Insurance contributions. This helps pay towards your UK state pension. When you register as unemployed, these NI contributions are also paid.
feasably, if one had zero hours contracts, you could end up not paying enough NI contributions.
Why would people have multiple jobs? Well for some, it means you can have jobs to fit into your lifestyle. For others, it’s the only work they can get. Like everything, some good, some bad.
I assume the US unemployment count is not much different from Canada. I know one of the stats here that occasionally gets mentioned (during economic downturns) is it’s the number of people *looking *for work. One concern is that they don’t really track the number of people who don’t count because they’ve given up looking. (There was also a lively discussion many many moons ago when they changed the count process, so that students looking for summer employment didn’t add to the total - “seasonally adjusted” I think.