It would have to be "needs to choose". Even English has a little bit of grammar. ;)
And in that grammar the verbs "need" and "dare" are semi-modals and can be used in the same way as regular modals in negative structures - "No one need choose" and "No one needs to choose" are equally correct. In some constructions the modal form is possibly even more common, e.g. - "No one need ever know".
It's perfectly correct (see my other comment) and means that nobody has to choose ("Nikt nie musi wybrać", perhaps?). Not exactly Duo's sentence, which is more "It's not necessary to choose", but not far off in general meaning.
Shania Twain has a song, "No one needs to know" while Damien Rice sings "No one need know", both meaning "No one has to know".
I think "no one need choose" can be correct, depending on the context. For example, if there is a claim that everyone has to choose something from a set of options, it would be correct to say "no one need choose" anything or "no one needs to choose." it is informal but it is not incorrect.
If 'nie trzeba' translates as 'one does not have to' as the hint suggests - and I don't see how it could translate that way in some contexts but not here - then 'one needn't choose' is another formulation that should be accepted for this sentence.