the relevant part of the lyrics is "夢見るのは乙女の大事な仕事でしょ？ 止まらないのは情熱的な牡羊座のせい！"
that seems to roughly translate to "Dreaming is a girl's important job, isn't it? It won't stop with this passionate Aries!"
(actually, looking at that again I possibly screwed up on the "it won't stop" part, but it definitely says Aries there.)
One thing I'm not quite sure on is that the importance of Aries (the constellation) isn't explicitly stated there, but it definitely seems to be referring to something living (because inanimate objects usually wouldn't be described as passionate) and I guess I don't have a problem with trusting Japanese people to understand.
"Jun-chan's birthday hasn't been officially published, but looking on the net I found she 's an Aries and has the 'Chinese milk vetch' birth flower. From there, if you think about it, Jun-chan's birthday seems to be April 8th"
This actually gives us some good info. On https://ja.wikipedia.org/wiki/誕生花#4.E6.9C.88 , it looks like the only match for that birth flower (in the Aries period) is April 8th [edit: March 22nd too]. If I can verify the birth flower, I think we know Jun's birthday (and only three years late).
Quick added note: if you run the wikipedia page through Google Translate, the matching flower comes up as "Rengesou"
In the fourth volume of the manga, there's pictures of all the characters with flowers. It looks like what's going on is that the flowers in the pictures look like birth flowers for the days everyone with confirmed birthdays was born on.
So, the flower in Jun's picture has been identified, and by googling some variations on the flower name, this poster seems to have decided that the 13th was a good choice [Edit: Actually, that's not right at all. They say they though it'd surely be the 13th, but the statistics say the 18th is more likely. They don't know what to make of this.]. They have graphs of appearance rates of dates in their searching on the page, and I'm not going to argue with those results.
So yeah, we're looking at April 8th, 13th, or 18th.... I think.
Okay, let me try to digest as much of this as I can.
1. We accept, on good faith, that Jun's star sign is Aries
Most character songs seem to be the character singing about themselves more often than not. If we take the lyric "this passionate Aries" as Jun referring to herself in third person, we can lock her into that birthday range.
2. Birthday flower image
That blog post also seems to list the birthdays/flowers for each of the other girls. However, I cross-checked the characters' birthdays to the image and to the birthday flower list on wikipedia, and it's not perfect. Like Ritsu's flower doesn't seem to come up, but Mugi's does. But I can just assume that the wikipedia page isn't exhaustive. However, I can at least accept that Jun's flower is definitely meant to indicate her birthday.
3. Jun's birthday is still somewhat disputed
From your second post, it looks like that blogger couldn't confirm the exact name of the flower? If there are a couple of possible varieties of the one flower, that of course sets up a few more possible birthdays. It looks like they've at least confirmed the species, since the three name variations they suggest (れんげ草, 蓮華草, レンゲ草) all point to the same species. But in the end, I've got no idea why three different name variations on the same flower would each yield a different set of birthdays. From point 2, I've accepted that the birthday flower list on the wikipedia page isn't exhaustive, but I still find it a bit strange that 3 variants of the same flower would crop up within a couple of weeks of each other.
Also, I'm not sure about what the graphs/data on that page actually represent, so I don't know how they conclude with 13th or 18th based off their tally. Not only that, but with that 'Rengesou' flower, you seem to suggest that March 22nd is also a possibility. Do you have any ideas as to why that date was left out of that blogger's analysis?
The initial dates for the analysis seem to have been chosen by means of "The hits for the three name variations "renge," "RENGE," and "renge (in kanji)" [I CAN ONLY ROMANISE IT SO MANY WAYS] on Google were used as "rengesou" [much easier than 'Chinese milk cetch'] birth flower candidates."
That's probably the dodgiest translation so far, but it hopefully tells us enough.... and that is that apparently the analysis wasn't exhaustive.... I don't know how I missed that.
FWIW, I assume that the shortened name was used here to get more results, although it may have skewed results because of Google's dark magic. I didn't actually notice that some of our days were missing, but if they're not common, it might make sense to leave them out.
For actual searching, "Among these [dates], only ones in Aries were counted [I think?]." And then.... there's something I'm not qute getting about search results and being from 100 of something.... not really sure, although.......
Crap. Maybe they weren't Google results? (I'd assume they were still, but mine seem wildly different).......... Oops. I gotta learn to read stuff. Just sorta assumed Google from earlier. (Or maybe it's just me using the wrong methodology - not sure on the exact searches done).
As far as the name variations for this specific flower, it's the same thing as the date finding stuff, it's "rengesou," "rengesou," and "RENGEsou." They're not different names, just different ways of writing it (perhaps I could've made that more clear).
THIS IS JUST RAISING MORE QUESTIONS THAN IT ANSWERS NOW. DAMN.
Perhaps we need to start at the start using only known facts: Aries and Rengesou (which I guess aren't confirmed, but they seem likely). Then we can verify all the data we see.
I guess we'd add up hits for searches from the more comprehensive date list and see where we get.
Actually.... there's not really a lot of point continuing now, unless it's just for fun.
We're not going to really get a guaranteed date, and just a likely date doesn't necessarily mean anything.
If we want something, lets just add something about April 8th being commonly used by fans (look at twitter - https://twitter.com/search?q=%23鈴木純生誕祭 , it's by far the most commonly used date among these tweets) to the trivia section (maybe with a little bit about this reasoning).