Cart before the horse.
In the case of the vast majority of Americans desiring to come to Japan to work, a degree is a part of the requirement for receiving a working visa. So it isn't a matter of whether an employer will accept you without one or not, since you can't get a visa to come do the job for them anyway. The degree is a requirement from the Japanese government, not the employer directly.
There are employers that don't require a degree. They use local foreigners who are here on spouse visas or working holiday visas (no degree requirement for those, and no working holiday visas for Americans). They are not desirable employers and generally have poor reputations.
Speaking from personal experience: you don't want to be a foreigner living in Japan without a degree. Whether you intend to live here or in the U.S. or anywhere, go get one if it all possible.