I am unsure how these squad mode abilities work. Do they allow the Battle Brothers to use their reactions to tactical advance/standard attack on their turn (in addition to their normal allotment of actions) or do they allow them to do those things *as* reactions during enemy turns, in lieu of say, dodging?

Good question. In our group we interpreted this the way that you can use those actions when it's your turn, sacrificing your reaction.

PS: Tactical advance grows quite strong once you've reached rank 5 and combine it with tactical spacing.