Qual das frases está correta? Tem alguma regra pro uso do just?

"I have just one" would be the normal, natural way of expressing it.

Há registros de "just" antes do ver e depois do sujeito, como nessas sentença:
"I just have one comment about the author's terminology." (forbes.com)
"I just have one question about the guide."  (washingtonpost.com)

Quando se quer enfatizar algo, "just" acompanha aquilo que se deseja enfatizar: "They're just great".
A questão é que "just" pode ser usado para representar vários nuances de significado na sentença. Dê uma olhada no Cambridge Dictionary para você ter uma ideia desses vários usos.
No caso específico de sua pergunta, tem que se ver o escopo de "just"; se for enfatizar "one", I have just one. Caso seja enfatizar a posse, por meio de "have", I just have one. Por exemplo, o juiz num tribunal pergunta para o promotor ou advogado: "Algo mais?", e o promotor/advogado responde:
A) I just have one more question, your honour
B) I have just one more question, your honour

Embora a) e b) pareça significar a mesma coisa, existe "a slightly difference" entre elas.
That's it!

Thanks for complementing the information. I thought about both ways, and the likelihood in terms of probability.
I mean, an example could be when talking about quantities (in terms of concrete things, not abstract) - example, if someone asks you how many cars or houses do you have? I have just one. Here 'I just have one' couldn´t be meaning what the speaker meant.

Still in terms of quantity "I have just one more question, Your Honour." could mean that (in terms of quantity, perhaps there was expected some 'quantity' of question, or at least not many.
Or the person (lawyer, D.A. Etc) would expect to make one or a couple of questions, not many.

Anyway, the word "more" rings of "an additional" component there, had he/she stated "I have just one" then definitely (or I think) the suitable way would be "I have just..." instead of "I just have..." (back to my 'quantity' theory).

Whereas with the sentence "I just have one..." would ring "só tenho uma..." as in:
I just have this problem with my attitude.
Eu só tenho esse problema da minha atitude.

I just have a little less time.
Eu, simplesmente, tenho um bocadinho menos de tempo disponível.

So, it suggests something like "eu só tenho/eu simplesmente/eu somente/tenho apenas...", etc, so it rings a bit on the rethorical side to me.

Of course, the example provided by Marcelo might be interchangeable (it goes with what the lawyer/D.A. Means at the time).
Whereas the case of a question about how many cars/houses you have/own it would be way less flexible.
English shades of grey!


That is, it´s similar to what happens in Portuguese:
"I have just one" eu tenho somente uma." (in case we know what was about, said previously, or asked).
"I just have one..." eu somente/apenas tenho uma... (certainly it [the sentence] would need more elaborating, hence the reticence points).