Can a Private pilot fly a company aircraft for business purposes if he is not being paid?
I'd say you'd be on thin ice with this. The principle is that private pilots cannot be compensated for their flying, with very few exceptions. That doesn't necessarily mean getting paid to fly, it can mean having the expenses of the flight given to you in some way. 

Say you take 3 friends on a trip, and your friends pay all the costs. You'd be in violation of FAR 61.113 because you haven't paid your share, in other words you've received compensation. You need to pay 1/4 of the costs of the trip. 

In another case you are going on a business trip and decide to fly yourself and your employer pays all the expenses. This is legit because flying is incidental and not the purpose of the trip. 

In the case you are stating you have no reason to be flying the airplane other than the flight itself, therefore flying for free means you are receiving compensation and a violation of FAR 61.113. 

Note other countries may vary, the UK is extremely similar to the US on this, as are most European countries, I can't speak for anywhere else.