Meanwhile, Eve Muirhead led the Scottish women to a 10-2 demolition of Latvia in just six ends to keep her campaign in good health.
McMillan has been on the back foot since he lost his opening two games on Saturday. He scored an extra end win over Russia on Sunday and then his team produced their best curling so far to beat the Czechs. They opened with a two in the first end and never looked back.
In the afternoon game, the Swiss took the early initiative, and although the Scots fought back to level the game at 4-4 in the sixth end, the Scots gave up a game-turning three shots in the seventh, when McMillan's counting stone lay open, allowing Switzerland a simple tap-out to score. Eventually, the Scots ran out of stones in the tenth. This result puts the Scottish men in sixth place, but crucially two wins behind the five teams who are tied for first place.
The Scottish women face winless Netherlands next.