Robina,
They don't suppress freedom of speech. They are a private entity with their rules which he most likely agreed to. They don't owe him a job. They can fire him for being rude to other employees.
Yep.
Freedom of Speech does not mean that one is free to say whatever one wishes, however one wishes, whenever one wishes ... For example, even the strongest advocates of the right will not do this: walk into the office of the head of the employing organization and engage in an hour-long abusive rant. Why? Firing is likely to be swift and without much of a legal leg to stand on. The guy is free to say whatever he likes in the privacy and comfort of his own home, or in a pub with friends, but care should always be taken in the workplace.
It seems most likely that the guy does not belong to a union, which typically would have an agreement with the employer to the effect that firing must be based on "just cause". If so, then his employment was "at will", which means that Google is free to fire him at any time, without warning, and with no obligation to give a reason. The only exception would be if his civil rights were violated, and I don't think even the guy is claiming that.
Most important: Regardless of how he views his rights and what sorts of claims he might bring forth on that basis, large companies like Google will have a Code or Ethics/Conduct that employees sign up to, and, in so doing might well waive certain legal rights. Some employers will even go beyond the workplace in such matters. Looking at Google's Code of Conduct, it appears that the guy violated certain parts, and that pretty much finishes his case (if he even had one to start with).
In my opinion, Google was absolutely right to kick out the fellow. Perhaps those who consider him a hero will offer him suitable and equivalent alternative employment, although my guess is that major employers will treat him with caution.