What does the halal label mean to non-muslims for non-meat foods? For example kosher berries or salad are checked more thoroughly for insects than usual as far as I know, so could be attractive to non-jews who don't want to accidentally eat insects.

Are there similar reasons to search out or avoid halal food for non-religious reasons?

I restrict this question to non-meat food because there is an existing question where the answers all focus on meat.
Since Halal food can't contain pork meat / by-products or alcohol, you'll see some ingredients substitutes (for example, Halal candy will not contain pork gelatin for sure; instead probably agar-agar will be used). So it might be advantageous if you can't consume alcohol or are allergic to pork.

As with any substitutions, sometimes you'll see a difference in texture / flavor, sometimes not.