Joining factions doesn't exclude you from anything in the game. You can, and should, join all four factions and complete their quest lines up to a point. The game makes it explicitly clear (with a pop up notification) when something you're about to do will make another faction hostile. You'll eventually encounter Danse regardless, but if you want to encounter him sooner wander around south of Graygarden toward Cambridge. You'll pick up a radio station (Military Frequency 95 or something like that) which will lead you to him.


He's not available as a full companion regardless until after completing the Reunions quest though.

After completing the Reunions Main Quest, the rest of the Brotherhood shows up, and Danse reports in for orders. Since you've probably also joined at that point, you go with him. After a talk with the Elder, you're able to have Danse as a companion. And no, I"m not sure what happens if you don't join before doing Reunions.

If you don't meet up with the Brotherhood of Steel prior to completing Reunions, a quest triggers as soon as you walk out of Fort Hagen and see the Prydwen. This quest just involves going and investigating the Prydwen, and you then get sent to the Cambridge Police Station. I've never actually gotten this quest though because I've always went to Cambridge prior to Reunions.