Q: What should be the affiliation of an institution's ex-employee in research publications?

Generally speaking, I don't think you need to use the term "ex-employee" in the affiliation. Researchers often move from one institution to another and the primary affiliation on the paper should always be the institution where the major chunk of the research for the paper was conducted. In this case, if the research was completed when your co-author was still in your institution, you can just add it as his/her affiliation in the paper. The institution where your co-author is currently employed should be mentioned either as a footnote or in the  correspondence address section, so that if a reader wants to contact him/her, they can do so.

However, having said that, the conventions may differ from one field of research to another. Therefore, it might be a good idea to find out the conventions in your area of research by asking your peers.

Related reading:

• How should a change of institute reflect in the author's affiliation?
• How should I list multiple affiliations in my paper?