Are they letting me go?

I don't think so...if your boss was going to fire you, he/she would just do it, they wouldn't bother making a lunch out of it. If anything, you're probably getting good news.

That's not the way most bosses would do it, but it is not outside the realm of possibilities. Who knows? The boss may be asking you if you want a permanent job.

In any case, there isn't anything you can do about it. Walk and talk confidently. If it's bad news, react calmly. If you think you are being unfairly treated, don't start spewing out poorly formed thoughts. Instead, ask a lot of questions to get as much of the story as you can. Then, pause. Ask the boss if you can discuss this with him/her again tomorrow morning.

That will give you time to gather your wits about you and put in a calm, reasoned defense. Ask the boss if he/she is willing to keep her/him mind open until then.

I had a similar situation. The boss will probably ask you to work for her directly. She will most likely also tell you not to tell the employment agency, so that her company will not have to pay to keep you as their own employee. If you do go behind the agency's back, don't ever expect them to get you another job though. Good luck.

It would be uncomfortable and unnecessary for a manager to take someone outside of the workplace for repremand or termination.

I think you are going to hear something worth listening to, so go to the meeting with a positive attitude and listen to what she has to say.

Why do you not like what you are doing? If the reason(s) are not fixable, then decline the invite and tell her why. If your feelings about the workplace can be fixed, then be prepared to ask for constructive feedback AFTER you have heard her out.

Go to lunch with her. She might be scoping you out to hire you on permanently. If you were going to get fired, I think the temp. agency would do it.

Have lunch with her. Be nice. Don't call her an ***.