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What is an appropriate word for a person who when criticized seems to think that they are completely unworthy in that particular category?
For example:
Me: "I wish you could be nicer when you speak to me."
Person: "Well I guess I am just a terrible person."
Another example:
Me: "I wish you could pay attention to me when I'm speaking to you."
Person: "Well I guess I am just the worst boyfriend to you."
Oversensitive, hypersensitive, easily offended, touchy or thin-skinned. Choosing one particular would depend on the occasion. I would describe the person as hypersensitive with a tendency to feigned self-punitive guilt-arousing behaviour in order to gain a moral advantage.
As for the particular examples you posted, the person also speaks with an air of being passive aggressive. With such over-dramatic and loaded responses, the person is trying to elicit guilt from you while making it seem as if they are being self-deprecating. While this doesn't strictly answer your posted question, it seems useful to point out based on your chosen examples.
The answer to this depends upon your interpretation of the the motives for their responses. The response is self-effacing, implying that they feel they are, as you say, unworthy, or inadequate or merely humble, but in my opinion, the person doesn't want to hear feed back from you and is responding this way to shut down the conversation. Words for that are evasive or distant.
But why are they that way? They could be insecure - lacking the self-confidence that they could positively change in response to your request. You could categorize that under defensive. They could be disinterested - unwilling to invest the emotional energy to maintain a give-and-take relationship. You could call that apathetic. Or they could be fed up with the relationship - perhaps they feel you are unresponsive to their requests. You could call this exasperated, spent, frustrated, or emotionally depleted. There could be many other explanations. Good luck with that person.
Someone who undervalues themself could be described as self-deprecatory. That is, they agree with the disapproval being expressed about their character, actions or motives.
