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This is really more of an extended comment, since I'm not sure how to give a definite answer. When $\dim M=1$, $c_1(M)>0$ if and only if $M= \mathbb{C}\mathbb{P}^1$ if and only if $M$ is simply connected, by the uniformization theorem. In the dimension $2$, things are more complicated. Certainly simply connected with $c_1 < 0$ exist. For example, any surface in $\mathbb{C}\mathbb{P}^3$ of degree $5$ or more will work. However, there are also plenty of nonsimply connected examples (products of curves of large genus, ball quotients...). Does this help?

Added Explanation: To explain where the examples are coming from and to answer your 2nd comment, let me explain I'm using Kodaira's embedding theorem to translate $c_1(M)<0$ to ampleness of the canonical bundle $K$. This condition is stable under products.
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This is really more of an extended comment, since I'm not how to give a definite answer. When $\dim M=1$, $c_1(M)>0$ if and only if $M= \mathbb{C}\mathbb{P}^1$ if and only if $M$ is simply connected, by the uniformization theorem. In the dimension $2$, things are more complicated. Certainly simply connected with $c_1 < 0$ exist. For example, any surface in $\mathbb{C}\mathbb{P}^3$ of degree $5$ or more will work. However, there are also plenty of nonsimply connected examples (products of curves of large genus, ball quotients...). Does this help?