Is it permissible to use eggs by placing them on the foot for treatment, based on the belief that it helps to mend a broken leg?

There is no objection to using food for treatment if it is not disgraced, according to the Almighty's saying: (It is He who created for you all that is in the earth), and this includes the general benefit unless there is evidence of prohibition. It is permissible to use henna with egg yolk to straighten or nourish hair if it is beneficial. The basic principle regarding things is permissibility, so the owner may dispose of them unless there is sound evidence of prohibition.