B767 Captain reported experiencing a tow bar failure during the push-back maneuver resulting in the aircraft rolling slightly due to the brakes not being set.

Narrative

Pushback at ZZZ ramp. Tug driver used standard terminology to begin push. Pushed back very slowly; then angled the jet is such a way that he could not straighten it on the taxi line. He stopped; pulled forward and tried again. Same result. He pulled forward one more time and told me he was having a bad night. He backed the jet again and got the tug to a point where it was almost 90 degrees out from the nose wheel and still not on centerline. We were close enough to the line that I could taxi; so I told him that if he could disconnect; we would taxi. He said he could not disconnect in that position. A mechanic came out to help with the problem. The mechanic hooked up and told me the tow bar had broken; but it was now disconnected. The parking brake was not engaged (no comm to ask me to set brakes). At this point I felt the jet rolling slowly forward. I manually depressed the brakes to stop the jet and told the mechanic that the parking brake was NOT set. He asked me to set brakes. To my knowledge; no person or equipment were affected. I asked him if there was any damage to the nose gear or his team/equipment; he said no. I cleared them off and taxied uneventfully. In total; the push took 15 minutes. Cause: Tug driver not proficient with pushback ops; inadequate communication between tug driver and mechanic when mechanic took over; did not communicate the parking brake was not set to him when he disconnected the tow bar. Suggestions: Mechanics; who are proficient at tug ops; should be doing it.