Should I quit my job over this?

VK1997
I have not been happy with my job for quite some time now. I've looked for new jobs but haven't been actively applying. I reached my breaking point on Friday and would like input based on what happened. (Sorry for the length).

Customer contacted us because an order had not shipped (a couple 8x10 prints). I looked into the order, see its still on time but the previous agent gave an incorrect estimated ship date (estimated for the day before). I apologized to the customer, upgraded shipping, and told them I would do everything I could to get it shipped that day. Customer was still pissed and after some back and forth, she told me the company sucked and to "go fuck yourself!" I responded immediately with "I will not tolerate being spoken to that way" before they decided to end the chat session.

Brought this to my managers attention. She scheduled a video call with me and my direct supervisor later that day. In the call she basically told me that what happened was my fault. She also said "I appreciate that you felt the need to stand up for yourself but you should've said it in a nicer way." I complained that something needs to change on the labs part because this is not even close to the first time this has happened to me or any agent for that matter. They just bend over backwards for the customer and tell us to "brush it off." She then asked if i would fire the client over this if i was in her shoes. I said "Absolutely. I would rather my employees feel respected and supported than maintain a business relationship with an unruly customer." She disagreed. The whole meeting ended without an actual apology for what happened and I ended the call after an hour and a half by stating that we needed to agree to disagree on the situation.

I feel that I should have complete respect and support from my superiors. They praise me often but the moment it mattered most, they were against me. I'm also worried because I currently make more here than what other, similar jobs offer.
yes, quit
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no, don't quit
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