As @raja11sep has already covered the part that Non-Reg language U Non-Regular language may or may not be Non Regular so I will cover only concatenation part .

Suppose L1 ,L2 and L be some non regular languages .

Let , L1 = L U {e}

L2 = L’ U {e}

(Since L is non regular so L’ is also non regular)

Now L1.L2=(L U {e}).(L’ U {e}) = sigma * i.e. a regular language

Hence concatenation of two non regular languages may or may not be non regular.

2.(L U {e}).(L’ U {e}) =LL’ U L U L’ U {epsilon} = L U L’ = sigma* (Complete Language).

3.{a^nb^n}.{c^nd^n}={a^nb^nc^md^m}

An example where, concatenation of two non regular language is non regular.Hence concatenation of two non regular languages may or may not be non regular.