Day One supports dynamic checklists as well as bullet points.
When you create an entry, tap the list icon once for a bullet point list, and tap it a second time to start a check list.
Once you have completed your check list, tap the Done icon. Now your check list is dynamic and you can tap to check off the items.
Type your list and then highlight it. Then you will see the formatting bar appear and you can choose the check list option.