It's not dangerous to drive without the antiswaybar connected, you just can't/shouldn't anyway drive it like a race car. Take it easy on curvy roads and it will be fine until you can get the antiswaybar reconnected with a replacement link.
1) No, the alignment would not have broken that link, which is prone to break anyway. It's a common problem that is quickly, easily, and inexpensively fixed by simply installing a replacement link. The broken link cannot be repaired.
A factory replacement link would be expensive from the dealer but they should be a dime-a-dozen or maybe even free used from people who have replaced theirs with quick disconnect antiswaybar links or Currie Antirock antiswaybars.
I'd place an ad in the Parts Wanted section here and at www.jeepforum.com
after looking through the ads to see if anyone has a pair for sale.
Or if you plan to offroad, it would be smart to just replace the broken link and the other unbroken link with a pair of quick-disconnect links from someplace like JKS. Which would probably be near the same cost as buying a new OE link from the dealer.