It is always "jĳ", or at least it can be. "Jĳ" is the full version of the word. There is however, an eroded version: "je".
This is similar to the relation between "you" and "ya", just that "ya" isn't usually accepted in the written language, while "je" is (though it's less common written language than in spoken language).
In both languages, the eroded versions can't be used when the word is stressed. Likewise, the full version has no such limitation. When in doubt, use the full version.
If you start comparing languages then everything looks like something else. If you read it "in Dutch" it sounds more like the "y" in "you" (je -> ye) so definitely easier to remember if you get confused with "je" in French (sounding more like "je" in "jet" or "Jennifer")
That's quite perceptive; many people don't seem to notice. There was a group of compound words that combined a form of "you" with a form of "people". I think the only one still more or less in use, if no longer current, is "jelui" - "je-lui" "you people".
The mnemonic doesn't quite make sense, to me, but then I guess that's not its purpose anyway. A mnemonic should just be memorable; making sense is not a requirement.
je/jij hebt - you have (sg)
hij heeft - he has
ze/zij heeft - she has
The confusing thing is that the pronoun u - the formal you - can take hebt or heeft.