Translation:I am sure that you are going to be fine.
I am a learner also, but I have seen the que translated as "to or that." De is of course "of." Whenever I see the word "seguro," I think of "secure," though that is not so accurate. So when I saw it, I read; "I am secure of that you are going to be well." It made sense to me.
De que is a conjunction and while que can be a conjunction (or relative pronoun) the test is can you switch out 'that' with "which"(identifies it as a pronoun). It's just a tip to help choose without getting to deep in grammar.
Here we definitely have a conjunction & we can't use 'which'. It's de que
Use "estoy segura de" if what follows is in the infinitive form or a pronouns such as "eso", "esto", "ello". "Estoy segura de haber encontrado el correcto" "Estoy segura de eso."
Use "estoy segura de que" if what follows is a phrase (or a verb in another form but the infinitive). "Estoy segura de que ella lo hizo."
(stolen from http://forum.wordreference.com/showthread.php?t=259785 )
I'm sorry you felt that way. Perhaps she came across as condescending or was too public with her correction - that definitely happens. I am determined to be grateful for correction by native speakers - I know that I hesitate to correct non-native speakers of English, as it is sometimes not appreciated. Other times they are quite grateful, as I would like to be. Darn eggshells! :D
It's true that it's wrong to say you're doing "good," but unfortunately it's so common these days that I definitely wouldn't bother correcting someone. The only reason the spanish speaker probably corrected you (I wasn't there of course, the person could just be an a$$hole) is because they realized you were learning and wanted to help you learn the proper way to respond :)
I think it is context, and Duo is thinking like a doctor. It does seem to me that this is the sort of thing a Mom might say, hopefully, when she lives her kid at the sitter's. Fine seems to speak of the child's well being, good seems to speak to the kid's behavior. There may be a distinction to be made here in Spanish, I don't know.
This is the sort of Duo thing that will drive you up the wall if you fret over it. Just put "fine" and move along . . . . . :)