The teenagers have their reasons (whether these are logical or you and you partner agree or disagree about them) It sounds a bit like a blackmail telling you that they won't see you while he is around. At least they are open to seeing you when he is not there. This indicates they that the conflict they've got is not with you but with him. If this had started most recently as you've indicated, it hints that something must of taken place (perhaps w/o your knowledge to let them change their minds)
Think back about the time when this took place- do you recall becoming more depressed/unhappy, angry, having different expectations of them (that may have been your partner's expectations, did you and your ex get in a scuffle, etc)
Behavior is usually purposeful. They have a reason for acting this way and you've as the mother and the adult will have to separate (how their behavior seems to hurt you or your partner) and redirect your energy onto hearing them out. If they say they don't know of any specific reason, then let them know that you appreciate their concern for you (they've told you that they don't like the way he treats you. Tell them that you've heard their feedback (sometimes people want to be acknowledged. As your kids they may feel undermined and even jealous now that you're in this relationship.
Ask them should they think of specific examples to please tell you or even email those to you so you can give it some further thought. This shows them that you are in fact willing to hear them out (it does not mean that you have to agree)
Another thing- you may not want to place the responsibility/burden onto them or anyone else for you emotional state and that of your partner. These are always going to be your kids whether this man is in your life or not at some point in the future. You do not want them to feel guilt or responsibility for you going to the hospital, etc. It is your perception and expectations of others (yours and of your partner) that causes you the stress and grief. It seems that the kids love you and have some concern (regardless of what is causing this shift in their behavior) You would want to appear to be the stronger one in this. The kids are very impressionable and ought to know that even if you do not understand or agree with them, you're giving it some thought and that meanwhile you're going to be OK. Otherwise, they may become alienated for fear of upsetting you further. Maybe that is why they've not come up with reasons (for fear of you becoming hurt)