minimal polynomial of a matrix B given minimal polynomial of $B^2$

Question

If we are given a minimal polynomial for a matrix $B^2$ can we deduce the minimal polynomial for $B$ $?$

Example:

if the minimal polynomial for $B^2$ is $m(\lambda) = \lambda^4$ then can we deduce the minimal polynomial for $B$ is $m(\lambda) = \lambda^8$

Edit

From my example would I be able to deduce $B$ has a Jordan normal form consisting of an $8\times8$ block with eigenvalue $0$