What happens if there's a company-wide no meeting day and I want to set a different day for my team?

When there's a company-wide no-meeting day set for an organization by their Clockwise admin, enabling a team no-meeting day will create an additional hold.

For example, if your organization has set Wednesdays as your company-wide NMD, and you set your team NMD as Thursday, Clockwise will honor both of these requests and will suggest neither of these days as available for scheduling within your organization. If both NMDs are set for Wednesday, Clockwise will honor the Wednesday as your hold day. 

Having more than one NMD enabled can cause some serious issues reserving Focus Time and providing week flexibility for meetings to move around in to accommodate maximum schedule flexibility in a cross-team environment.

It is possible to disable the team no-meeting day for a team for the individual via the settings as discussed here.