Your partner might be in complete denial about what they're doing to themselves, and if your gentle nudges to spring for greener meals or go for a run with you are received poorly, then your partner likely isn't thinking long term about how their decisions could affect them—or they just don't care.
What they should definitely care about either way? You.
If encouraging them to join you on your wellness journey isn't helping, try something along these lines:
I love you, and I'm very scared that you might get sick or—if worse comes to worst—that I will lose you. It can be really devastating to see you (eating so poorly, ignoring exercise, or whatever other unhealthy habits they have).
I need for you to not only care about what you are doing to yourself but to care about the effect it's having on me (and the family, if applicable).
I need to know that you care enough about me to do what you need to do to stay healthy.
Hopefully your partner just doesn't recognize the way their unhealthy behavior affects you in the present and could affect you in the future. If after this conversation they still refuse to change, then it's important to recognize what that means: The real problem is not just that your partner doesn't care about themselves—it's that they don't care about you and your life together enough to make a change.