I propose that this question should be undeleted:
https://mathoverflow.net/questions/304492/reduced-ring-with-all-non-prime-ideals-finitely-generated

It was deleted by the OP despite having a substantial answer by Keith Kearnes.  Keith has re-asked the question and re-posted his answer at
https://mathoverflow.net/questions/304533/reduced-ring-with-all-non-prime-ideals-finitely-generated