Is this usage of "aren't" proper English?

I don't think there's anything wrong with it. It's a contraction of Are you not going to go outside?, which is perfectly acceptable; the addition of the negation changes the expectation conveyed by the sentence to imply that the speaker thought the subject was going to go outside. This is useful, and it's as meaningful to ask about an event not occurring as it is to ask about it occurring.

I suspect that your wife's problem is really with the sound of the word "aren't", especially when pronounced as one syllable. (Ahr-ənt sounds passably okay, arnt sounds hickish and suggestive of the dreaded ain't.)

To start a question with aren't is perfectly acceptable.

Using the negation in some questions can give to the question a different meaning, and the question could be interpreted as a suggestion.

Aren't you going inside? — The suggestion is to go inside.
Didn't you mean to call Frank? - The suggestion is to call Frank.

To express a question when not knowing someone's will I would use the more familiar

Are you going inside?

The problem with

Are you not going inside?

is the ambiguity of the single-word-answer "No" along with a double-negative. "No you are not" or "No you are not 'not going' "?

I am no scholar but I am a Hillbilly. We normally just say something like "Ain't ya goin' on in?" which is just horrible in most circumstances.

Contractions such as "aren't", for example, should only be used in casual/informal speech/writing. So, the answer is that it depends on the context.