The "continuity of rights" is a straight translation of the French continuité des droits. That expression means that the social security (i.e. health insurance) rights of a person do not expire after a period of time, i.e. are continuous, uninterrupted in time or here in space. Every person steadily living or working in France has the right for social protection all along their life.
The form you had to sign (Convention de stage / Internship agreement) is a prerequisite for the student to be allowed to be an intern at your company.
Here is how I understand both choices:
#1 The host organization (you) provides an insurance that supplements the medical care refund granted under French law (in case it doesn't already cover the actual costs.)
#2 The host organization insurance doesn't provide extra medical coverage so the intern won't be refunded more than what the French social security covers in such a situation.
You might have a look to this page to get more information about your specific case, as both the status of the intern and the target country matter, depending on the agreements between the local and the remote health insurance providers.
It looks like there are several variations of this form, depending on the university, the time or whatever factor.
Here is another one from the very same Sorbonne:
SOCIAL WELFARE PROTECTION FROM THE HOST ORGANIZATION:
By checking the appropriate box below, the host organization indicates whether it provides health insurance coverage to the intern
under local law:
YES: This coverage is in addition to the maintenance abroad of rights granted under French law.
NO: coverage is thus exclusively provided by the maintenance abroad of the rights granted under the French student coverage framework).
The French text reads:
Protection sociale issue de l’organisme d’accueil
En cochant la case appropriée, l’organisme d’accueil indique ci-après s’il fournit une protection Maladie au stagiaire, en vertu du droit local :
• OUI : cette protection s’ajoute au maintien, à l’étranger, des droits issus du droit français
• NON : la protection découle alors exclusivement du maintien, à l’étranger, des droits issus du régime français étudiant).
It confirms my initial interpretation above.