Short Answer: No
In the Mohs scale of mineral hardness, teeth have a hardness of 5, where as steel has a hardness between 4 to 4.5. So the teeth are slightly harder.
If the shield were made only of steel, the zombie squirrels could scratch it. Given enough scratches (like, a bajillion), they should be able to eventually bore a hole through it.
I think it is fair to assume that vibranium is at least as hard as titanium. With a hardness of 6, titanium is safe from zombie squirrels.
Since his shield has the ultra resilient vibranium in it; it is unlikely the zombie squirrels can scratch it. Their teeth, bones, and everything else would wear away long before the shield showed even a scratch.
On the other hand. You have to consider that Squirrel Girl took out Thanos.
And Thanos shattered Captain America's shield.
So what we have here is a classic paper/rock/scissors scenario.
Squirrel Girl beats Thanos, Thanos beats Captain America's shield, Captain America beats Squirrel Girl.