Does this character rejoin the squad?

The answer is yes, Spoiler:

but it will be after the assault on the Citadel by Cerberus.

Eventually, Ashley/Kaidan will be fully healed but will be recruited by Udina to be a bodyguard for the Councilors; Ashley/Kaidan is unaware that Udina and Cerberus are working together to lead the Councilors into a trap so Udina can take control of the Council. This puts Shepard and Ashley/Kaidan in conflict; Shepard must then convince Ashley/Kaidan into realizing the truth about what's going on and turning on Udina instead. If Shepard cannot persuade Ashley/Kaidan to back off, then Shepard can kill Ashley/Kaidan. If not, a squad member will shoot her/him.

If Ashley/Kaidan lives past this event, Shepard has the option of either recruiting her/him back onto the Normandy and into the squad, or of leaving her/him behind to assist with the War Effort.

Source

Yes, they will after a certain event. Spoiler:

After Cerberus attacks the Citadel