The person isn't actually asking for permission but it means that it's the most natural thing to cry and it means why I shouldn't cry. your translation means "sensiz günler beni ağlatmasın mı"?
And yours means "sensiz günler için ağlayamaz mıyım?"
The problem here is that a word-for-word translation is impossible. We just have to pick a loose translation that gives the same feeling. I think mine leaves less room for misunderstanding.