I remember for a couple of days after my first 'that,' I really did worry I might be sick, and I didn't tell a soul.
Literal
First [attributive-の] 'that' [attributive-の] day [topic-は], certainly illness [quotative-か][quotative-と] two-days-[about-ほど] anyone [to-even-にも] not-say-[ren'yōkei-ず] was-worried memory [subject-が] exists-[polite-ます].
A delicate, roundabout reference to the speaker's first menstrual period. アレ ('that') in katakana is a common coy euphemism for menstruation and other topics euphemized in Japanese conversation. 思い悩む ('to worry deeply, to be troubled over') is a compound of 思う ('think') + 悩む ('be troubled'). ~ず is the classical negative stem-linker ('not X-ing and'). ~ほど 'about, to the extent of' here marks approximate duration.