Would you hire a burnt out engineer?

That depends on exactly what the candidate means by "burnt out" but it is a red flag that warrants asking some deep and probing questions. If it's just shorthand for "I was tired of my old job", fine, that's not a problem, probably. If it's actually incapacitating or the candidate has a history of such, well, that would be very concerning. I've been in the throes of long-term burnout before and I know very well how hard it is to get anything done when that happens.

I can't say it would disqualify them out right, but it would definitely be a major red flag. If you are a solid candidate then I'd be asking questions to understand how you got burnt out and why.

Being burnt generally means one of a couple things (there are others but this is where I'd start wanting to understand):

1. You can't manage expectations so you overload yourself. e.g. you can't say no

2. Your manager was poor and drove people into burn out.

And to be clear, I wouldn't be asking you if you just can't say no, I'd be testing you in the interview to see if you can be disagreeable. If you can't be disagreeable then you'll just do the same thing for me and I'd be replacing you again shortly if I hired you. So it would be a pass.

If instead it is #2 or some deviation of the same, then it'd be less of a red flag, but still concerning to make sure you were good by the time you start on my team.