Let $M$ and $N$ be two real square matrices of size $p+q$. The matrix $M$ is nonsingular. The matrix $N$ has the following block structure, where $A$ is a $q{\times}p$ matrix. $N = \left(\begin{array}{cc} \text{O}_{p,p} & \text{O}_{p,q} \\ A & \text{Id}_{q} \end{array}\right)$. I would like to conclude that the algebraic multiplicity of zero as an eigenvalue of the product matrix $MN$ is $p$. (The geometric multiplicity of the zero eigenvalue is preserved as $p$ from $N$ to $MN$.) I am looking for arguments to prove (or disprove) that conclusion.

Take $p=q=1$, let $M = \begin{pmatrix} 0 & 1 \newline 1 & 0 \end{pmatrix}$ and let $N = \begin{pmatrix} 0 & 0 \newline 0 & 1 \end{pmatrix}$. Then the algebraic multiplicity of the zero eigenvalue in $MN = \begin{pmatrix} 0 & 1 \newline 0 & 0 \end{pmatrix}$ is $2$.

It's not true. Consider the case ($p = 2,\ q=1$) $N = \pmatrix{0 & 0 & 0\cr 0 & 0 & 0\cr a & b & 1\cr}$. The characteristic polynomial of $MN$ is ${t}^{3}- \left( am_{{13}}+bm_{{23}}+m_{{33}} \right) {t}^{2}$, and $a m_{13} + b m_{23} + m_{33}$ could easily be 0, resulting in an algebraic multiplicity of 3.