The weird part at the time was that I had no conscious fear of losing. I knew the matchmaking system was going to make me lose 50% of the time no matter what I did, and I had placed in gold, which was much better than I'd expected. Yet still there was this unpleasant churning sensation associated with 'find game' that wasn't there with the campaign, and it kept getting worse.
The first breakthrough realisation for me was that I wasn't scared of losing, I was scared of not winning. That might sound daft, but what I mean is: I noticed I was far more reluctant to 'find game' after a win than after a loss. Playing a human opponent was very stressful and adrenal, and emerging with a win was a satisfying high that I was, in effect, reluctant to 'bet' on the outcome of the next game.
The next puzzle was why playing a human opponent was so stressful compared to the single player campaign. Again, I wasn't consciously aware of any ego being at stake (it literally never crossed my mind what my opponent might think of me as a player, beyond wanting to appear well-mannered) so it seemed to make no sense.
Eventually it clicked: the campaign is designed to be beaten. It's a story in which I'm the hero: I'm supposed to prevail in the end. 1v1 games aren't like that. In 1v1 I have no special status; there is no underlying narrative - and yet the outcome is still all down to me. It's not about ego; it's not about someone else knowing I've lost. It's fear of taking responsibility for an action whose outcome is entirely unknown.
That's why so much of the advice in the original thread was along the lines of 'play to lose' or 'do wacky strategies' or 'play some 2v2/3v3 etc': all these techniques are ways to either predetermine the likely outcome or dilute individual responsibility. Bizarre but true: we are happy to deliberately lower our chances of winning, just so we know they're low before we start!
While these tactics work in the short term, as soon as you start 'trying properly' again the anxiety will return. Fortunately it's possible to tackle the problem more constructively by shifting your objectives to things you can control. You are not clicking that button to see if you win or lose, you are clicking that button so you can practice staying on top of your spawn larvae, or try out a new lair timing, or whatever. Take a plan into the game and define 'winning' as the execution of that plan. Your opponent is just giving you feedback on how solid the plan itself is.
When you nail that way of thinking, it's amazing how positive your games start to feel. Instead of being angry or disappointed because a mis-micro lost you the game, you can walk away thinking "Hmm, well, that plan put me in a position to win/extend the game if I had microed correctly - cool!"
Edit: what if, as someone asked, your micro is consistently getting in the way and preventing you evaluating the strength of your plan? This is a great question because it illustrates how easy it is to successfully sidestep the 'Will I win this next game?' issue and then proceed to make the same mistake in a subtly different way. Evaluating the strength of your plan is not your objective. Executing your plan is your objective. Being able to evaluate the strength of a plan is, like winning the game, a fringe benefit of focusing on execution. If you make your objective the evaluation of a plan you are not able to competently execute, you will become frustrated because you'll be thinking "Well, I didn't learn anything that game."
So there you have it:
Step 1: Understand that it is your inability to predict or gauge the likely outcome of a 1v1 in advance, combined with your undiluted responsibility, that generates the anxiety. It's not ego, and it's not a character flaw. As a species we are literally genetically programmed to find responsibility without control inherently terrifying and unpleasant.
Step 2: Take control in positive ways. Don't play to lose, throw games away with cheesy strategies, or hide in the crowd of 3v3 games. Set yourself attainable in-game goals, and let your win-rate do its own thing.
Step 3: Enjoyment!