Could heartbreak cause dysthymia?

Most definitely: That would not be at all unusual.

Heartbreak: It is certainly possible to be sad for a long time after heartbreak. If depressed mood, low energy, low self-esteem, poor appetite or overeating, difficulty concentrating, sleep disturbance, and feelings of hopelessness last 2 years or more, the person meets criteria for persistent depressive disorder (what dysthymia has been changed to in the new dsm-5 diagnostic manual. Therapy can still help.

Yes: Dysthymia is a type of depression, and who wouldn't feel some depression as well as a myriad of other feelings after a breakup? These feelings can be explored meaningfully in psychotherapy, if you so desire.