Thanks to @Yosemite Stan, we have a counterexample, and actually many of them: Pick a projective variety $Z\subset \mathbb{P}^n$ with some odd-cohomology and non-vanishing $c_1(Z)=m>0.$  

Denoting by $\widetilde{Z}$ its image via the Veronese map
$\mathbb{P}^n\rightarrow\mathbb{P^{\binom{n+m}{m}-1}}$, define $X\subset \mathbb{C}^{\binom{n+m}{m}}$ to be the cone over $\widetilde{Z}$.
There is a natural resolution $Y$ of $X$ given by the closure of its preimage under the blowup at $0\subset \mathbb{C}^{\binom{n+m}{m}}.$ 
As blowup at $0$ is $O(-1)\rightarrow \mathbb{P^{\binom{n+m}{m}-1}},$ 
$Y$ is precisely equal to $O(-1)|_\widetilde{Z} \cong O(-m)|_Z,$ hence has $c_1=-m+c_1(Z)=0,$ and cohomology equal to $Z,$ hence not supported in even degrees only. 

In particular, choosing the quartic 3-fold $Z$, we have $b_3(Z)=60$ and $c_1(Z)=1,$ and $X=V(z_0^4+\dots+z_4^4)$ is an isolated singularity and (trivially) a complete intersection.