There are two different ways to go about using logic to determine which respondent is allowed to pick a time or enter their personal information in an Ethnio screener:
Option 1) Qualifying Questions
Setup one or more qualifying questions on the first page of your screener. That will give you the most robust control over which participants are allowed to pick a time or enter their contact info. Please note you cannot route respondents to multiple schedules using this type of logic. Read more about qualifying questions here.
Option 2) Drill Down
Use the Drill Down question type for a single thread of more simple questions than above. Keep in mind the series of questions all have to be of the type "drill down." Then the last one could be "Scheduling." Lots more here on how drill down questions work.
There's no correct way to setup your recruiting screener. The question types, filtering options, and design are all customizable and up to you.