There are three views/controls that will let you display one cell per day:
1. Navigator (a small monthly calendar that usually serves as a date picker)
http://doc.daypilot.org/navigator/
2. Monthly event calendar (a similar format but it allows you to insert items/events in the day cells):
http://doc.daypilot.org/month/
3. Scheduler (it displays days on the horizontal axis for multiple resources):
http://doc.daypilot.org/scheduler/
All these components will allow you to select days by clicking on them. This user action will fire TimeRangeSelected event. The rest is in your hands - you can mark the clicked days as selected using a custom CSS class, insert an event with details (saying "selected" or something similar), and allow actions depending on the user role (clear the selection, select other days, etc.).
You can also mark the forbidden days using BeforeCellRender event (use a custom color or CSS class):
http://doc.daypilot.org/scheduler/cell-customization/