Yes, there is.
It's called Kouyohoken. It's pretty similar to what you described. It's part of a package that bundles unemployment insurance, pension, and national health insurance together.
The percentage is about 80% or so. It last for about 90 days. You get it within a month if you're fired or if your contract finishes. Otherwise it's a 3 month wait. Pregnant mothers get ******.
Many Gaijin don't have this option since most Eikaiwa schools would prefer screwing their staff and save themselves a bundle by breaking the law. The law is too lazy to check all the schools let alone 1 I bet.
I pushed my old company into giving me national insurance and then later at the end of my contract used the unemployment office to put pressure on my old company to pay for my past dues on unemployment insurance that they felt I wouldn't want.
There is no discrimiation beyond pregnanat women and the fact that you must be employed full time or up to a certain amount of hours that are close to being full time working hours.
One Eikaiwa school told me when I went for an interview that "You don't want national health insurance, a pension, and unemployment insurance" ... I basically called the guy a dumb ****. I didn't get the job needless to say since I knew too much about the lawful and rightful system that they hide from gaijin.
errrrrr ...