Should I ask for positive/negative feedback on a chat system?

I'm building a chat to use on my admin dashboard, where my clients interact with the company and ask for support. On this windows, I'm thinking about using a feedback button, to see how my clients are rating this system.

This is the screen I'm using (under development):

Whit this in mind, I have 2 main doubts:

Should I place the feedback right at the beginning of the conversation, as soon as the chat start? Or after some time/amount of messages?

Also, is it ok to ask only for positive feedback? Or should I keep the "Dislike" button so clients are able to "Dislike"?

The main purpose with this is to evaluate the chat system itself, and what they think about this new method, not the conversation or the topic discussed on the conversation.