Didn't bat an eye:
To not display even a hint of an emotional response, such as consternation, annoyance, sadness, joy, etc. (The Free Dictionary)
Usually you can take this to mean that they didn't actually notice the difference from usual, and hence didn't react to it.
They were wearing dog costumes at the party, but people didn't notice any change from the usual, because they were used to that sort of thing.
I wouldn't say it was very usual to add "over it" since it's already implied by the idiom itself. I don't see any issue in adding this in though, it just adds further clarity to the sentence.
As pointed out by @jonathonjo, bat an eyelid is also idiomatic and holds an identical meaning. Infact, I believe this is where the phrase originates and has just been cut-down over the years.