If my boyfriend has sickle cell anemia and I don't have neither the trait or disease, will our child end up having it?

No. If your boyfriend has sickle cell anemia (2 abnormal genes) and you have 2 normal genes, then your children will all have sickle cell trait but none will have the disease. You need to be sure you have no other blood abnormalities like thalassemia or hemoglobin c trait which can be totally asymptomatic; this would change the answer significantly!
No. The child is pretty much guaranteed to have the sickle trait, but it will not cause any significant trouble by itself.
No. The children will not have sickle disease but they will become a carrier, sickle trait.