I will suggest Tawk.to for all free looking requirements and if the data is so critical which is shared in the chat, then you may need to setup and use some thing in your hosting itself like phpchat or some other paid softwares mentioned above.
I used to use chatra which were really good but for those really good features I had to pay so I recently moved to .tawk.to which is free and is really good, it also has paid features which look really good.
I have tried so many different live chat systems, but the one that I do enjoy using and like is tawk.to it is a great system, and also has an ios app in which goes to my phone when someone requests live chat. I also get notifications when I receive visitors to my website and I know which pages of the website the guest is on.
I have strong opinions about livechat software since I clock 50-60 hours a week in it.
Previously in this thread I stated that I liked Livechatinc.com's chat product. I kinda still do, but I think it has declined significantly over the years in reliability, while adding no major features. At a price tag of $40/month per seat that just seems unacceptable to me. I've even actually had some features go missing that I used to use, only to find they've moved further down the pay wall.
I just switched us over to tawk.to and so far I'm pretty pleased, especially at the price. The only thing killing me about tawk.to so far is that there's no auto-pickup to the livechat. You have to pick up a ringing chat like a phone. If I'm at my desk and marked available, I just want it to throw me into the chat without the process of picking each one up individually.