Does this character rejoin the squad?

Yes, they will after a certain event. Spoiler:

Yes. To do so, you MUST have done all of the following:

1. Save the ORIGINAL Council in Mass Effect 1.

   • If you let the Council die, then in ME2 and ME3 the NEW Council feels that you would put human interests before galactic concerns (something Cerberus would do).

2. Visited them at the hospital all 3 times.

   • While comatose.
   • Awake but injured.
   • On their feet awaiting discharged.

3. (Optional) Romance them.

   • Stay faithful to them in ME2.
   • If pursued new love interest in ME2, apologize to them or tell them why you left them.

4. (Optional) Give them a "get well soon" gift.

5. If BOTH, or either, Thane or Kirrahe are alive and spoke to both of them. Then when Kai Leng tries to kill the Salarian Councilor, Thane or Kirrahe will jump in and save the councilor. If the Salarian Councilor was saved by either of them, Udina has no "evidence" of his own to frame you with.

6. Depending on the dialogue you choose when confronting Udina, Kaidan/Ashley, and the turian and asari councilors - take the interrupt.

As far as if Ashley or Kaiden return to the Normandy, they don't have to but yet, there is the option to get them back. Just keep playing the story, I can't say any more than that without spoilers. What I can say though, is you won't be able to miss the conversation that determines this.