There is no question of "hospitals don't want people without green cards", but hospitals generally require employees to have legal work permits in the United States when recruiting people.

The laboratory provides clinical inspection services.

Many hospitals don't have clinical laboratories, but some independent laboratories in the society provide clinical laboratory services for hospitals. These laboratories are usually called laboratories in America.

H 1B identity and work

If you have the status of H 1B, you must have a job, otherwise you won't get H 1B and you won't get H 1B to find a job.

ASCP exam

If you can take the ASCP exam, there is still great hope for finding a job in the United States.

H 1B application

However, it is not that you use H 1B to find a job, but that after you find a job, your employer will sponsor you to apply for H 1B, regardless of whether you have a green card or not.