I have 100 names set up like this:
First Name Last Name * First Name Last Name * First Name Last Name * First Name Last Name * (etc.)
I have set my text boxes to justify left and right.
Is there a way to set rules, e.g. to avoid "*" at the start of a line?
Your question is not very clear, can you share a screen shot? Are you using different Paragraph Styles?
I'd try a GREP style that finds [any word][space]asterisk - \w * and applies a No Break Character Style. Might work.
Do a find/change for the * and change it to a non-breaking space followed by the *.