I'll start with a few basics.
To get an invention patent (there are also design patents to protect an ornamental design, I'm ignoring those now), the invention itself needs to be novel and non-obvious (over basically all prior public disclosures).
That means, any disclosure of how the invention works voids your ability to obtain a patent. In principle. There are a few exceptions, namely, if one of the inventors disclosed the invention, you can still file for a US-patent for one year (grace period for own disclosures). However, a European patent for example wouldn't be possible anymore. (There are other countries with a grace period, so some international protection is possible, but to a lesser extent).
So you can tell the company about everything that has already been published (oral disclosures in front of a few people not bound by secrecy do count). And that's probably what they are asking about. They can then judge if they want to work with you to file a patent. The risk is relatively low because your disclosures are prior art for their patent applications even if they did file without your consent ("stole your invention").
If you think there are still parts of the invention that are unknown to the public, the other answers already discussed a few points, but I'll reiterate.
The universitys technology office might be helpful and you might be obligated to include them anyways
An NDA would be "nice to have", but they might not sign one because it is a strong weapon against them coming up with your invention on their own. It's in most cases not a good business decision to sign one with an inventor.
Try to not discuss specifics before having a deal.
If your invention is stolen, you can sue the stealing party, but, jokes on you, you'll have to prove it. I recently spoke with an experienced attorney in Germany who said he never had a case where it was even tried.
A last point, all this sounds worse than it is. Unless they can easily include your invention in an existing product, they probably want your expertise and having a patent additionally would be nice. Most firms don't steal your invention.