Both of these are fine to me. You could also say 'Why don't we go out for a meal this evening?'

The first one is okay

The first one is fine, either written or spoken. The second one sounds better spoken, because if it were written, there should be "if" after "about."

As a native English speaker, neither of these feel correct. I don't know the linguistics for why. "How about going out for a meal this evening?" makes sense, is understandable, and I think would be totally acceptable in a casual setting. It's okay, but not formal. "How about we go out?" is awkward, but I don't think it's incorrect. It's awkward because of the repeated 'ou' sounds. Instead, I would choose one of these: "Do you want to go out for dinner this evening?" (I don't know why, but I never hear people say 'a meal' -- it's always coffee, breakfast, lunch, dinner, etc.) (Also this would definitely have romantic connotations.) "How about we go get something to eat?" (This would be used more for friends. It's way less romantic.) "What about going out for lunch?" (This would be used in the context of offering a suggestion. It still has the repeated 'ou' sound, but it's alright.)

Whoops, answered this in the wrong spot! Sorry!