Ha-ha. The schools have definitely not taken into account some regional and local diversity into their coffee chat concept.
Here are a few tips:
1. Business Casual means slacks (that means pants that are not jeans but not black suit pants either - something in the middle and a shirt. A nice Polo shirt can work if it works but a dress shirt would be preferrable for most). No tie. You can wear a sports coat if it is cold or a sweater over the shirt. You would wear shoes (not sports shoes or sneakers but black/brown leather shoes)
2. Business Casual refers to clothing you would wear to work but not something you would be uncomfortable in - i.e. no tie and no suit - those tend to be very formal. You can definitely wear them to the meet but you would stand out a bit too much and that may be a misunderstood move.
3. You want to stick to coffee, juice, sparkling water, or tea unless the person you are meeting orders a beer first. I am sure some cultures may be more susceptible to alcohol (i.e. germany or france) while others may not appreciate it. The main reason to being conservative is that you don't want to make your chat buddy uncomfortable or judge you too early in the meeting/conversation. Some people don't drink for example. You can work a drink into your conversation - e.g. I have had a long day and I can really use a beer - would you like one too?
4. In terms of paying - you would split it in half usually. These bend to be school-sponsored and but the right thing is to depart equals or to kindly offer to pick up the tab since the other person has graciously spent their time with you. I would not make a big deal out of it but would work into a conversation such as "This has been a great evening and I really appreciate the opportunity to meet you, can I pick up the tab?" and then they can say - don't worry about it, it is all on school's dime, which is great but you don't want to come off assuming that or imposing again.
In other words - you want to be a polite guest who would want to be invited again