That's a myth. While being an employee, there's no stability whatsoever. You can get fired any time - get laid off, become a victim of corporate politics, or get fired for whatever other reason. Just last month, someone I know got fired on the spot after 15 years at a company for not reporting some personal stuff between other employees.. something he wasn't involved with, at all. I'm sure everyone else heard stories of people being thrown out after 10-20-30 years of loyal service to the company for bullshit reasons.
At the same time, when you run a business, you can't ever get fired, and everything depends on you. Whether you make money, or lose money - it's for the most part, your doing (market circumstances nonwithstanding).
That's why running a business is a lot more of a stable job than working for the man.