Some parents tell their girls (say aged 5 and upwards) to wear the headscarf when out for example in a non Muslim school in a non Muslim country. Is this right in Islam or should it only be enforced at her first menses and onset of puberty?
There isn't any religious obligation on children to observe hijab:
رفع القلم عن ثلاث عن النائم حتى يستيقظ وعن الصغير حتى يكبر وعن المجنون حتى يعقل أو يفيق
The pen has been lifted from three: From the sleeper until he wakes up, from the minor until he grows up, and from the insane until he comes back to his senses or recovers.
But there also isn't any prohibition on them doing so.
Muslim parents usually do educate and train children on religious matters while they are young so that it is easier for them when they reach the age of accountability.
ياأيها الذين آمنوا قوا أنفسكم وأهليكم نارا
O you who have believed, protect yourselves and your families from the Fire
And training them about religious obligations is part of protecting them:
وهذا يدل على أن علينا تعليم أولادنا وأهلينا الدين والخير وما لا يستغنى عنه من الآداب ، وهو مثل قوله تعالى : وأمر أهلك بالصلاة واصطبر عليها ونحو قوله تعالى للنبي صلى الله عليه وسلم : وأنذر عشيرتك الأقربين
This [verse 66:6] is evidence that we have the responsibility of educating our children and family members on religion, good acts and proper manners. And other examples of this include: "And enjoin prayer upon your family and be steadfast therein" [20:132] and the saying of Allah to the Prophet, peace be upon him: "And warn your closest kindred" [26:214]
يعني : مروهم بالخير وانهوهم عن الشر وعلموهم وأدبوهم تقوهم بذلك نارا
That is: command them to do good and forbid them from evil and educate them and teach them good manners and by this save them from the fire.
كلكم راع وكلكم مسئول عن رعيته ... والرجل راع على أهل بيته وهو مسئول عن رعيته والمرأة راعية على أهل بيت زوجها وولده وهي مسئولة عنهم
Surely! Everyone of you is a guardian (shepherd) and is responsible for his charges ... a man is the guardian of his family (household) and is responsible for his subjects; a woman is the guardian of her husband's home and of his children and is responsible for them.