Let $R$ be a ring with unit. $a\in R$ is called unit regular if there exists a unit $u\in R$ such that $a=aua$. A ring is called unit regular if every element of $R$ is unit regular. Examples of unit regular rings include division rings, boolean rings and strongly regular (von Newmann) ring. Finite unit regular rings include $\mathbb{Z}_p$ where $p$ is prime and these have no proper ideals.
Question: Are there examples of finite unit regular rings that have proper ideals?
Yes, for example $F_2\times F_2$ where $F_2$ is the field of two elements.
Any finite, nonsimple semisimple ring is an example. So a product of multiple matrix rings of any size over any finite field is going to be an example.