No, it will not break anything.
If/when a companion is called away from the settlement on a quest, they will be removed as leader for the settlement, you'll get a msg saying that companion X don't want to be leader any more. That pauses the city build as it was. No new plots will be built. However, the plots already there will keep on developing.
Thus, when you assign a new leader, might be the same, to the settlement, the city build will have progressed a bit even without a leader present. To level a city you need in-game time to elapse, enough plots of a certain level, and resources. Some of that develops even without a leader present, but there might be a shortage of beds or other stuff as there's no new plots being built to meet the demands of a growing settlement. But there's a safe guard for that as well. If there's more than two (the default setting) settlers unemployed, the recruitment [beacon] shuts down until more job openings [plots] have been created.