There is the Dubai Medical College. I accepts students from all nationalities but you need to know English to be able to study there. It's also pretty expensive. I think the annual tuition fee is like 100,000. I don't think you can work and study at the same time because you'll need to attend classes and how would you do that if you work?
Alternately, you can try to do a paramedics (Emergency Medical Technician) course whose advertisement I saw in Kabayan newspaper. The course consists of "9 weekends of lectures and 3 months of skills training in hospital and field internship with Dubai ambulance." Contact the International Health and Safety Training Center (
http://www.ihstcuae.com) for information if you are interested.