Simple question really: Noli quaerere quid vobis tua patria facere potest, sed quid facere tuae patriae potes.
Does this say "Ask not what your country can do for you, but what you can do for your country."?
I'm not sure if you're asking for alternative translations or just want to know if your translation is correct, but if the former, here's a looser, but I think more idiomatic, version:
Non quid patria tibi sed quid tu patriae prosis consulendum est.
This is using prosum + dat. "be of service to" to capture the "do for" of the English, and instead of an imperative "ask" I used the gerundive consulendum est "it is to be considered, one should consider".
@cnread correctly points out that the verbs inside the indirect questions must be subjunctive. The two instances of "quid" introduce indirect questions.
I also made the verbs plural. I think this is more consistent with your use of "vobis", and perhaps with the original Kennedy quote as well.
Here is a revised version:
Nolite quaerere quid facere pro vobis patria possit, sed quid facere pro patria vos possitis.
As for the other part, your reading is correct. "Ask not what your country can do for you, but what you can do for your country."