This article jumps to way too big a conclusion. In the very beginning of this article, they cite the Sleep Foundation. I went to the article they cited from, here: How to Train Yourself to Sleep on Your Back. A complete article backed by a sleep medicine physician that undermines Mayo Clinic's story. They say that back sleeping may be unhealthy for some, if you have problems like heartburn, sleep apnea or you want to reduce snoring. But the rest of the article explains and encourages sleeping on your back.
Another article from the Sleep Foundation states, "The best position depends on your unique needs, but back or side sleeping is considered better than stomach sleeping." ( What Are the Best Positions for Sleeping? )
Furthermore, the "sleep specialist" quoted in the article that makes this claim has her studies in psychiatry, and pulmonary medicine (lung disorders). She studies sleep, but her focus is mostly on narcolepsy. Lois E. Krahn, M.D. - Doctors and Medical Staff - Mayo Clinic
Even if she is credible, she is jumping to too big a claim, when there is plenty of other research saying that back sleeping is okay, and further, stomach sleeping is worse. Mayo Clinic could have easily said that back sleeping isn't a good idea for some. Saying it is straight out unhealthy is misleading and false according to The Sleep Foundation and the BBC.
If you want more information with less exaggerated claims, the BBC explains it the best: Which is the best sleeping position?