D3 Upgrade

First, I am really sorry if there is another (or million of topics) addressing this particular question, I couldn't find one that could really help me.

My question is:

I am registered as a D3 non-rev at AA, have flown stand-by in business class a few times, now my company will pay for my ticket (normal economy ticket) and I am wondering if as a D3 I would be able to ask for an upgrade to business class at the counter if there are seats available (since I am - or de company is paying way more that I would if traveling stand-by).
Really hope I was able to explain myself well. It is a really long travel with 2 connections (both ways), 3 different countries (really not too close) and one 15 hour flight and another 8.5 hour flight, an upgrade would really make a difference there.

Thanks for any help in advance, and again sorry if there other posts that talk specifically about this.