Cara
Core beliefs are hard to find because they usually are attached to some trauma and because they have a built in protection designed to keep them from being lost since they control our thinking. A good therapist will pick at you as a way of drawing your attention to them since so often you can't do it on your own. Anger is a sign you are in the area. It is often a way of stopping you from going there which is bad since it also gets rid of the anxiety with out addressing the problem. That built in protection I just mentioned. So what makes you angry can be a finger board pointing in the right direction.
If I was your therapist I would send you back into your past looking at anything traumatic related to health. And every time you tried to avoid some thought I would push you to seriously try to cut through the protection. To accept it and look at it to see if it needs challenging.
Core beliefs can be suggestive and not related to personal experience, what creates them may only be associated.
In the case of health it is usually some trauma caused by a serious condition happening to someone near or dear to you, but it can also be a near call that you fear happening again. At this point I have to say that accepting the thought and challenging it is the only answer even if the answer is that yes you have a good chance of getting what ever you fear.
Better the devil you know than one you don't.
Examples of situations that cause core beliefs are things like a person you know getting a disease they should not have, possibly because they are too young or their is no history of it. But don't rule out associated thoughts. A person getting killed in a stupid car accident can affect your thoughts on other ways to have your health go down hill. The association in this case is the fact that they are all unnecessary or unlikely enough to sneak up and you need to be on guard at the first sign, hence the anxiety which we know is a survival technique to protect us. Which leads to the most common way of dealing with it. Getting mad as a form of challenge.
If I was your therapist I would be pushing you to open and look at anything traumatic in your past as far back as you can remember, and not ruling out things only closely related. The farther back a trauma is the more time it has to be built on with other traumas and the stronger is the belief. Blank spots are a sign you are close and don't want to go there. Work on them, they will open unless they are very very traumatic in which case you will still know the reason for the core belief.
That in itself is enough to challenge and change it.
This is only one step in the CBT changing of thought patterns. A therapist may recommend a temporary use of medication to help open blockages and get past the built in protection. This is rare though and needs supervision.
Davit.