Can a Genjutsu awaken Mangekyou Sharingan? It was said that Itachi's genjutsu was so good that both Kakashi and Naruto either felt the after effects of it as if it were reality, or never even realized in the first place that they were in a genjutsu. 

So, can a strong enough genjutsu be used on someone to make them witness the "death" of someone close to them and thus make the user under the genjutsu awaken his or her Mangekyou Sharingan?
Not possible, Sasuke didn't awaken his Mangekyou Sharingan after being battered by Itachi's Genjutsu. Itachi kept on replaying how he killed both Sasuke's parents. The Mangekyou can only be awaken when the user suffer from trauma after witnessing the death of someone close to them. With the user normally having to witness or experience this first hand not by Genjutsu. For further details please refer to Mangekyo Sharingan