As well as the actual definitions, these are slang expressions. A regular restaurant or cafe would simply have "men/women" or "gentlemen/ladies" to indicate the restrooms. So the speaker is saying that a place which has restrooms marked as "dudes/chicks" is too casual a place to discuss something as serious as the ending of their marriage.

Guys and girls, usually adolescents or young adults. Being single has nothing to do with it. It is mostly used by the aforementioned age groups but not exclusively.

chicks = young single girls dudes = young single men

In modern usage, the word "dude" has become gender neutral. What it means is that he hangs out with the other "dudes" too much. Maybe you can find somebody more articulate, more mature, and with a better earning potential. flirting with girls though most girls dont like being called chicks....:))