Hi Peter,
I'd only be speculating as to what benefit he might get by signing you up for the cover (if indeed he did sign you up, and did gain a benefit).
Perhaps if you had an accident - his insurance policy may not have covered that (and probably came with a much cheaper premium because of that) - so he needed you to have the insurance to safeguard him in case you had an accident and the injured third party made a claim against him or the company because they were better targets to sue in terms of their assets. As I say, that's just pure speculation.
Without you having a contract it will be difficult for you to find a legal basis to get your $360 back (given that you have verbally agreed to him getting the policy on your behalf). If you have any trouble doing this, I recommend that you at least try to obtain evidence of the insurance policy from the company. This could be useful for two reasons:
1) if he has taken out a policy, it should be valid for 12 months or so - you might be able to use this in your next self-employment role;
2) if he cannot produce any evidence of the cover, you'll have more of an argument as to why you should be refunded the money. You can talk to your local community legal centre or QPILCH about this further (they will give you a free face to face or phone legal consult - just google "community legal centres Queensland" ) - as you might find that you have grounds for a "minor civil dispute" in QCAT if you agreed for him to buy cover of a certain length (e.g. 12 months) and he just took your money without getting the cover.
As I say, it's probably best you talk this option through with one of those services because you're likely to need quite a bit of advice before considering this option, because the facts are quite novel (and you would need to invest a lot of time and effort).
In terms of going forward - it sounds like you might want to try self-employment again in the future (I did too when I arrived in Australia from overseas). A good resource I'd recommend is the "Fair Work Australia" website - they should have some good resources on the rights of independent contractors - you will want to read up as much as you can on your rights and obligations as an independent contractor in Australia (they are treated a lot differently to employees here).
I understand that there some peak bodies for independent contractors that you can join - although they are also likely to have great resources for you on their website.
Good luck with your Aussie adventure.