And if you think this is an unreasonable idea... a quick look at health insurance quotes shows me I can get a decent comprehensive health care plan for under $100/month. That's for a plan that I'm almost certainly going to use year-to-year. I've probably had a good half-dozen covered health expenses for my family in the past year, for example.
In comparison, unemployment insurance is - ideally - for catastrophic situations, which (hopefully) turn up infrequently. You just keep socking it away as an investment until such point as you need to start drawing on it, or cash it in entirely. Think of it as a 401(k) that you can access without penalty if you're unemployed for more than 2 weeks, and there you have it. Just another specialized investment fund.