Q: Is there any time that it is permissable to administer the Eucharist to a Non-Catholic, and if so what are the conditions that have to be met?

A: Yes there are such situations. The Vatican has issued guidelines concerning this.

For Eastern Orthodox, they can receive the Eucharist if they request them and are properly disposed (Code of Canon Law 844:3).

In cases of danger of death or grave necessity, Catholic ministers may (by the permission of their conference of bishops) give the Eucharist to Protestant Christians who (a) cannot approach a minister of their own community, (b) who ask for the sacraments, (c) who share the Church’s faith concerning the Eucharist, and (d) are properly disposed to receiving it (CIC 844:4).

Under no circumstances can the Eucharist be given to a non-Christian (an unbaptized person).