Will this recipe turn out if you uses all egg whites and 0% plain yogurt? (That's all I have in my fridge right now!)

I don't think the yogurt will be a problem, but missing 4 egg yolks would cause you to lose a lot of richness. And egg whites are drying, so the balance of 4 whites and no yolks would cause the flan to be very dry. But as I'm writing this, I see that you posted the question over a week ago. What happened?

I expect it won't be as creamy (custard without eggs = ??!?!). I don't think the non-fat yogurt is that big of a deal. I used 1%. But the only way you'll know for sure is to try it out! Please come back and share your findings.

Your chance of success is much better with the correct ingredients. No-fat yogurt and egg white will lack fat, which carries flavor. You might be able to make up for the fat deficit by adding butter, but why muck around with figuring this out when you could get the proper ingredients?

- Baking is very tricky so chancing ingredients in a big way can have unexpected results.

Two options A. Go to the store and the the proper ingredient or B. Make it and find out.