Have a large fatty pocket, maybe lipoma, that's leaking brownish fluid. Can I squeeze it to get the fluid out?

lipomas don't leak out brownish fluid unless it's possibly infected which would be rare. It sounds like you probably have some type of cyst or something else there. I would recommend you get it evaluated by a board certified plastic surgeon.Best Regards,Dr. Gartner

Im not sure what you are referring to. Do you have photos? that could help us immensely. Lipomas don't usually leak any fluid unless you have actually broken the capsule they are in and now the fat is draining out, but that is usually gold in color. Is there redness? you might have an abscess or infection and you are just expressing inflammatory fluid or serous fluid. A picture would help a lot here. 
This answer has been solicited without seeing this patient and cannot be held as true medical advice, but only opinion.The information provided should not be relied upon as a substitute for consultations with a qualified health professional.