If I Counter a Break Attack during a Move Action, can I continue moving after resolving the Counter?

Yes.

Countering gives you an Attack Action. This would normally end the current Action. However, Break Attacks are an exception to this rule. The Break Attack Rules on PG. 44 specifically say:

“Unlike other Actions, a Break Attack and resulting Actions will not end the movement of the figure that is moving. Any remaining movement may be finished after the Break Attack ends”