If the show eventually answers this question, just let me know, but don't tell me the answer. I don't want to be spoiled.  So, with that in mind...

Why did Healer confess to Yeong Shin as his alias, Bong Soo?  He did that KNOWING that she had romantic feelings for his secret identity "Healer".  Not only that, but he knows that he himself has romantic feelings for her.  But, from his confession, it seems that he would prefer to be accepted by her as Bong Soo, for some reason.  But I can't even begin to fathom why that might be.  Does the show eventually address this?

For the record, that confession scene.... really the sweetest thing I've seen in a drama in a while.  

"Are you confessing right now?" (A look of genuine disbelief mixed with concern. Min Young's eyes are so sincere here)
"Mm" (nodding his head, cool as can be)
"Sincerely?" (her concern very subtly switching inward, almost suspicious but curious about why this is suddenly happening)

^ I almost fell off my couch due to 'feelz'.  I LOVE that one little extra word they wrote in: "Sincerely?"  She's taken aback, but her focus is on really understanding her friend's feelings while trying to avoid misunderstanding.  Although I may not understand the Why of this scene, I really really like that the confession happened so unexpectedly and in an unexpected way.  It was very romantic and touching.

Hello ArisJaggamar, 

I can tell why he did that, but I think you don't want me to do so. Don't worry the show will tell you why he did that!

Watch on! More "feelz" to come. 

P.S: I agree. It was one of the best confession scenes I've ever seen. I skipped a heartbeat. 
Thanks, Lumiere.
I DID come up with a theory that's sort of simple: he needs to keep his true identity a secret for now, but doesn't want to wait to be with her.  But for me, I still don't fully grasp why he so desperately wants to keep his identity secret from her.  Perhaps to protect her?  To protect himself?  I'm dying for him to just lay himself bare to her.  She's so honest and open and loves being alive; the complete opposite of him, of course.  I know she'll protect his secret...

ANYWAYS, I don't want to overanalyze this!  I will take your suggestion, Lumiere, and just wait until I learn the real answers!
@arvisjaggamar healer is my fav action romance drama so just cant resist replying

u will know later why he wants to protect his identity . but still to ur question that why did he confess as bong soo is not something that will be explained later . i think its something as u said that he doesnt want to wait for her . he is there with her and so want to take the opportunity . arent they both cute
Yes, @amitspiky!  Healer has only gotten better and better since I first posted this.  I'm on Episode 19, and i am so sad that I have reached the end.  At this point, I'm sure I am right: he couldn't reveal his true identity to her at the time, but still wanted to be with her.  Really, just extremely adorable.
@ArvisJaggamar 

I think you got your answer by now, but here's my two cents on it. 

I think that Jung Ho never considered revealing his Healer identity to her for many reasons. The most important one being that it would put her in danger. He was a wanted criminal, and knowing his identity will entangle her in all sort of things. Ultimately, he was trying to keep her safe. 

His confession as Bong Soo- which is one of my all time favorite confessions- was also a promise. He promised her that if she wanted, he would live the way she wanted him to be "for a long time, carefully, by her side." In other words, he was telling her that he was ready to give up being the outlaw HEALER and become a normal person for her, for their relationship to be safe and normal. Here, it's important to note that being the 'HEALER' wasn't just Jong Ho's job; it was who he is and everything he'd got at that time. Come to think of it, he didn't make his confession as Bong Soo to start with. He didn't keep Bong Soo's usual antics. There was no stuttering, no awkwardness, no clumisness. At that moment, and perhaps for the first time in years, Jong Ho interacted with another human being as Jong Ho. Sure, Yeong Shin only knew Bong Soo at that time, but for Jong Ho, the man who lived practically without an identity as Healer and had virtually no real connection with the outside world (ahjumma was basically a sound in his head), that was the time when he wanted to be himself and shed down everything else for Young Shin. 

In my opinion, Healer wasn't just the love story of Young Shin and Jong Ho; it was also Jong Ho's journey of finding himself and returning to normal life. He did basically live in an underground cave away from humanity as Healer. It was his love for Young Shin that brought him back to light, to being himself, and living a normal life. Knowing that, we can assume that he never confessed as HEALER (and Yong Shin found out he's Healer on her own) because deep down he was never  HEALER. Healer was more of a mask he put on, as he had buried the real Jong Ho long before after he was abandoned by everyone he once considered as famtily (father died, mother abandoned him, his friends and teachers pitied him, his mentor abandoned him too...). Ultimately, Jong Ho had done the same, he abandoned himself under the mask of Healer.  
I'm thinking it was because 'Healer' was the 'real' him where as Bong Soo was a fictional character he had to create in order to hide his real identity so it would have felt like a false confession coming from someone made up. That's what i'm thinking anyway. I can't think of any other reason to be honest 
@Lumiere

Thanks for that response!  The idea of the confession being a promise is wonderful, and I hadn't truly considered that.  I really need to re-watch this.