Maximal Ideals in the Ring of Complex Entire Functions

No.

Let $f(z)=1/\Gamma(z)$ where $\Gamma$ is the gamma function. Then $f$ vanishes at $0$, $-1,-2,\ldots$ and nowhere else. For integers $m$ let $f_m(z)=f(z+m)$. Then $f_m$ vanishes at $-m,-m-1,\ldots$. Let $I$ be the ideal of the ring of the entire functions generated by the $f_m$. Then $I$ is proper, since otherwise finitely many of the $f_m$ would generate an ideal containing the constant function $1$. But any finite set of $f_m$ have a common zero, so this doesn't hold. So $I\subseteq J$ for some maximal ideal $J$ (using the standard Zorn's lemma argument). But $J\ne M_\lambda$ for any $\lambda$, as for each $\lambda$ there is some $m$ with $f_m(\lambda)\ne0$, and $f_m\in J$.

If I remember correctly from my student days, if D is a connected domain in the complex plane, then the ring O(D) of holomorphic functions on D has the property that every finitely generated ideal is actually principal, but there are ideas tat cannot be finitely generated (such as in Robin's answer above)