I have a Jacobian matrix, and after subbing in the equilibria $0,0$ I get
$$[-1 \ \ 0 \\ 0 \ \ \ \ 0]$$.
The eigenvalues are $0$ and $-1$. I believe that because of this there will be a line of fixed points which will be stable. My question is how do I find the eigenvectors and where will the line be?
The eigenvector of $\left[ \matrix{-1 & 0\cr 0 & 0\cr}\right]$ for eigenvalue $0$ is $\pmatrix{0\cr 1\cr}$.
There may or may not be a line (or curve) of equilibrium points. It depends on more than just the Jacobian at one equilibrium point. You need to look at the equations: if the system is $$ \eqalign{\dot{x} = f(x,y)\cr \dot{y} = g(x,y)\cr}$$ you're looking for solutions of $f(x,y)=0, g(x,y)=0$.