BD,
This is a good opportunity for me to share some things that I've only recently figured out and I'll explain my situation a little bit. My entire seduction career I've only really ever taken on fuck-buddies because most girls would fall short of what it took for me to date them seriously, so, for a long time I neglected the "grey" area: a girl was either my fuckbuddy or my monogamous girlfriend and there was no in-between.
Recently, however, I've started to dive into MLTRs which, essentially, entails you caring and vibing with a girl but not being committed to her: you treat her like a girlfriend, you can take her on dates, you can hang out with her but you're not committed to her. This is the area in seduction I'm living in right now and exactly how you're describing your situation it seems like that's what you have: an MLTR.
So, you fuck this girl, she invests a lot and you make no explicit remarks that she's going to be your girlfriend; all fine and normal. That's exactly how this situation should go for around 3 months - right around the 3 month mark she'd start to push hard to figure out "Where are we going?" The point to take a way is that you can legitimately care about a girl AND still not commit to her, just for future reference: I'll segue back to this in a minute.
Anyway, in this case, it could have been one of two things; she realized you were never going to be serious about her so she felt it was a waste to tell you about another guy or she still wanted to be serious with you and felt like telling you would push you away (a la "If he knows I'm fucking another guy he might not want me anymore"). But, if she flat out said "I don't think it's right to talk about this kind of stuff" then she's either baiting you into asking or she's following one of the two lines of thinking I just mentioned. This is kind of where that 1-3 month mark becomes important, if she did this early on (significantly short of 3 months) then I'd guess she's just trying to bait you into giving her more attention or she's trying to remain "pure" in your eyes so you might end up dating her, etc.
In either case, I'd respond to the texts as follows:
Her: I feel like shit today

Me: Uh-oh, sounds like a day of bed rest and ice cream!
You're acknowledging how she's feeling and making it fun for her, she'll probably read that text and smile. Basically, it doesn't matter why she feels like shit, it matters that you're showing her you understand how she's feeling at that moment. Opposed to:
Her: I feel like shit today

Me: Why's that?
Her: I don't think it's right to talk to you about this kind of stuff.
I'd say...Me: That's totally fine. It's a bummer that you're feeling under the weather. Let me know if there's anything I can do to help
You: Seriously, what's the matter? <-- at this point it feels like you're forcing things
Her: I said no.
You: Why won't you let me show you that I care!
In any case - if you were trying to reduce tension then you show her that you care by understanding her... not by understanding why she's sick or feeling shitty that day. Seguing things back, you should be showing a girl that you care about HER early on - women want to feel totally desired by strong men and when you can show her that then she falls into your lap; the girl I'm seeing now knows I desire HER, not just her body, not just her vagina, etc. and she's extremely attached to me as a result so, when I ask for nudes I get no resistance, when I want to fuck her I don't get resistance in a typical way (she actually does have an issue where sex can be a little painful at the moment!) and if I ask her for investment she has no issues doing something for me - all because I desire her and I set that precedent early on.
Yeah, though, you're right: You don't want to ignore her (which isn't what I advocated for) but you don't want to ask her questions to get her to feel comfortable talking about other guys, it's between her and another guy and if she wants to tell you she'll do so voluntarily. Ignoring is a lot different than being upbeat yet closed-off. My texts above are very upbeat and cause no friction while showing her you care/understand/making it fun. So, yeah, early on you want to avoid this a bit more but if she becomes an MLTR or something a bit more serious than a FB you can be a bit more lax but in any event you want to minimize how open you seem to talking about her/your relationship(s).
Best advice I can give you in this situation, regarding texting, is figure out what her intention is with the text she's sending and then respond to that intention; if she voluntarily texts you that she feels like shit then her intention is to find support from you and I crafted my text around that. Actually, now that I'm finishing up this response and thinking about it; if she refused to tell you what happened more than once then she was probably adamant that it wasn't your business. In my text above, after she refused the first time I didn't push, I back-tracked and showed her I care so that if a) I wasn't intruding on something she really didn't want to talk about and b) if she wanted me to care, I showed her that I cared.
-Richard