Can a breast cyst be painless and solid or is it always squishy and painful?

A lump that is "squishy and painful" anywhere is more likely an abscess which is a pocket of pus that has developed as an infection has progressed. This may or may not be associated with another process such as a benign cyst or cancerous nodule but usually comes from a superficial skin infection like an ingrown hair or pimple. It needs to drain, either spontaneously or with help. See a doctor.