OP - A couple of things do strike me - what did he exactly do during the years age 18 - 25 as just listing being in an undergraduate program in engineering is not enough on a resume. Any employer is going to want to know more details. If he does have even just the graduate credits, but not completed masters degree that is another issue and more explainable. If he had a health issue, it might just be easily referenced with the correct wording in a cover letter.
I would suggest paying for him to see a career counselor who could review his resume with him, help him develop a two or more kinds of resumes for various jobs as well as a suitable cover letter. The career counselor might also chat with him about what he wants to be doing, what his skills are and what his interests are. And perhaps one-on-one with him information will be shared which you may not know about in terms of how is feeling about his job search and why?
There is some reason why a 25 year old, engineering grad is not getting off of "dead center" in looking for a job so my next suggestion would be for him to have a full physical including at least a screening for depression, anxiety which might not be so evident to you living with him.
He is doing himself no favors by just working at a menial,low level job. For whatever reason could he have lost his confidence being turned down by a number of job applications? For whatever reason, an employer needs to see that he is following up on his undergraduate career interest - and perhaps taking a course as an Adult Learner at Maryland or GWU for example which could be applied to a masters degree would serve the basis of (1) renewing his self-esteem as being able to do the work in his field and (2) more importantly give him "something active" to list on a resume with EDUCATION at the top under Continuing Education and/or in a cover letter.
It is important for him to be treated as a full adult in your home with home responsibilities and a timeline on his progress on finding a job or finding out the reason for lack there of is key to helping him have a future. STEM education can be a real slog and some schools as where my husband just retired from do not have an easy option if someone finds it is not for him/her. So a final word if your son really does not want to pursue it, let that also be an option, but again menial jobs will not serve him well then either. He would need with the job counselor advice on networking, how to build upon his engineering background in looking at other fields from teaching to business to law for example and see how best to find that first job.
Finally, if you really would like an empty nest, you might let him know what you feel you and DH could afford if he did find a job new area in terms of helping him move and set up a first apartment.