If a person that works with children has a national background check and it show there is nothing there. Later after hired 60 days in asked to do a fingerprint and it shows it was an arrest and it is 16 yrs ago and it it is not a crime against children is this grounds for termination?

An employer in the United States does not need "grounds" for termination. This is an "at will" country, which means that an employee may be terminated at any time for any reason or no reason at all. The courts do not involve themselves in second-guessing employers' termination decisions.