Prove that the language of squares is not regular using homomorphism
As illustrated by other answer and comments there seems to be no easy way just using morphisms. Still it is a good observation that simple closure properties can be of help.
In your case I would suggest intersection with regular languages as a tool: $L\cap 0^*10^*1 = \{0^n10^n1 \mid n\ge 0\}$ whch some recognize immediately as non-regular. 

Alternatively one could include morphisms, inverse morphisms and intersection with regular languages (or, equivalently, finite state transductions). With these regularity preserving operations $L$ can be transformed in to $ \{a^nb^n \mid n\ge 0\}$, mother of all non-regular examples.