If I start/buy/run a company with my money, I can certainly hire and fire anyone I want, for any reason or no reason at all. Wouldn't want to have it any other way.
Now, whether that is a smart decision is another question.
Rarely is anyone fired for "no reason at all" or even "no warning at all". You might not know the reason, or caught the warning, and again, that is mgmt and the company's fault for not have the correct evaluation/feedback polices.