...if I am on holidays or on a RDO I am acting as a private citizen so QF cannot have any say over what I do.
I'm certainly not a lawyer, but I've recently had to write an internet use policy for a small business that employs Facebook loving teenagers and this is my understanding of the issues.
It's not so much a question of what you do, but what you publish. Posting on the internet is legally considered publishing. If you wrote and published a book slagging off Qantas while an employee, I assume they'd be terminating your employment. Tweeting etc is legally the same concept, particularly if you are identifiable by name and therefore as an employee. It's certainly not like criticising them in a private letter to an individual, although many people seem to see Tweets / Facebook in this way.
Qantas has no say over what you do in that they can't stop you doing it, but they can choose to dismiss you if you breach the terms of your employment. You could maybe chase an unfair dismissal claim based on the 'harsh, unjust or unreasonable' clause in the Fair Work Act (s385)(b) but I don't know how you'd go.
From what I can see, your only legislated workplace rights involving complaints about the company are the right to complain "to a person or body having the capacity under a
workplace law to seek compliance with that law or a
workplace instrument;" (s341(1)(c).
If you have agreed to abide by a code of conduct as a condition of employment, your employment is conditional on abiding by that code of conduct. I haven't read the Qantas Standards of Conduct policy referred to above, but I imagine it has a clause about not publishing nasty things about the company, whether on or off duty? Most of them do. Legally, Qantas is not trying to silence you as such, but excercizing their right to dismiss you for conduct breaching their code. No-one has a 'right' to be employed by a particular organization, particularly if they have breached an agreed term of employment.
If they considered your publication defamatory they could press charges, but otherwise once you were dismissed there wouldn't be anything they could do to silence you.
The Act does not seem to specify rules for codes of conduct, but my guess would be that unless a requirement is discriminatory, unlawful or harsh/unjust/unreasonable, an employer can specify pretty much anything they choose. At the end of the day, being a bastard to work for is not in itself unlawful.