Both expressions express wishes and expectations for health, and the specific expressions can be chosen according to the situation and language sense. In daily life, people usually use "keep healthy" to express their wishes for the health of others.
"Healthy and permanent" means hoping that someone's health will remain stable, free from diseases and health problems, and this state will continue.
"Stay healthy forever" expresses a stronger desire that someone's health can be maintained forever and will never be troubled by disease.