Who should I specify as my affiliation if re-writing an undergraduate paper? Years ago, I, as well as a team of other engineering students, conducted some pretty solid research for an undergrad project. We didn't end up commercializing it or anything, but I believe it would be a great fit for an upcoming conference.

I am no longer affiliated with the university where the research was originally conducted. The original paper was horrible, and needs to be re-written (I've been published before and am familiar with writing styles and the process of submitting a paper, despite not having an adviser).

Can (or should/must) I use the university as my (and my team's) affiliation? What does this actually mean? Does the university have to know I'm publishing a paper with their name on it? If so, who typically signs off on this?

I've tried to find an answer online and have come across this answer, but I'm not sure if it applies since I am not currently enrolled as a student, and because the paper is being re-written.
Your affiliation is listed so that you can be identified/contacted. Thus, your affiliation should be listed as your current employer.

Given that your university provided resources to conduct the research, you should list them in the acknowledgements section.

You should probably invite whoever supervised your project to co-author the paper. You should also invite your fellow team members to co-author the paper too.