My students compliment my teaching and say they now approach the subject with newfound enthusiasm. Can I ask them to put this in writing?
You may certainly encourage your students to provide evaluations that accurately reflect their opinion of your teaching. You may encourage them to provide detailed information rather than just the bare minimum. You may also explain to them how the evaluations are used by administrators.  What you must not do is to encourage them to make you look good by giving evaluations that don't accurately reflect their true opinions.