My students compliment my teaching and say they now approach the subject with newfound enthusiasm. Can I ask them to put this in writing? I'm a beginner instructor teaching undergraduates, and I seem to have their attention and am working hard to inspire them.  Recently several of them have told me that they now approach the subject with a newfound enthusiasm because of me.  

Anonymous teacher evaluations will probably be an important factor for me to remain employed; yet, I doubt that young undergrads know the significance of the evaluations, and I wonder whether they just circle any ratings on the evaluations form, just so they can get it over and done with as quickly as possible.

So, I am wondering whether I could make an announcement to the students at the end of the semester, when they have to evaluate me, that if they felt I've done a good job, this is their chance to support me and put it writing on their evaluations and to be as specific as possible.  

Is this ok to ask of the students?
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.