Icons from face-down Walk cards do count for "no X icons" Highlight of the Day bonus requirements.

This is supported by the following from page 15:

• "A player can count the number of walks they have taken by looking for icons, which are at the top of any Walk cards left face-down, and Walk tokens, which should have been placed each time they chose to activate a Walk card by turning it face-up."
• For "X" icons: "Bonus requirements showing an 'X' through an icon mean that a player must not have any of that particular icon activated on any of their cards in their itinerary up until that card to score the bonus."

The rulebook consistently refers to icons "activated on any of their cards," and for Walk cards, the icons at the top of face-down cards are considered activated for the purposes of bonus requirements. This is further clarified in the context of Walk icons, where both face-down icons and Walk tokens (from face-up cards) are counted.

Summary:
Icons from both face-down Walk cards and activated face-up cards are counted when checking "no X icons" requirements for Highlight of the Day bonuses.

Source: page 15

Source: Base Rules (pp. 10, 15, 6, 7)