While obviously this can be very specific, not just in disciplines of engineering, but also perhaps to individual departments and groups of different institutions with different focuses, I'll try to give a somewhat general answer.
First authorship is definitely important - it generally shows you "did the work". An important consideration for a hiring committee is how well your research direction fits into their focus. Your first author publications show "your" own work, and that you can do research. Generally these would be impressive and in high ranking publications, to show impact and recognition from other researchers.
Your "non first author" papers are either from collaborations with other researchers, or from supervision. Having good collaborations (perhaps with other institutions) shows a good ability to interact with colleagues and others in the field, which means you can likely attract funding. An important part of an academic job is to attract funding. Collaborating with others shows that others were (presumably) prepared to work with you, and there were results.
As a supervisor though, the quality of your students' output is also important. At least in my experience of engineering, the nearer the end your name is, often the more senior you are. For the avoidance of doubt, I am not referring to the addition of senior grant holders who had no input - let's presume we have 3 or 4 authors on a paper. The first one may be a PhD student. The second might be a collaborating PhD student. The third may be a RA or more senior PhD student, who helps the fourth (the supervisor) with supervision and advising students.
To give an anecdotal example (also in engineering), I started off writing papers with only myself and advisor as authors. After a few months, I found myself helping out some new-start students, and helped them get their own first publications. I ended up as second (of a few) authors. This continued and other students would approach for help and advice, and I'd be happy to assist them. As this happened more, I obviously have less active input in the specifics of the work, more helping to plan the research, and helping to confirm findings and challenge their own interpretation of the results. This pushes you down the author list towards the supervisor position.
I suggest that you want a healthy balance of work - good, high impact first-author papers to show your skills and expertise and research ability, combined with collaboration work, showing work with colleagues at other universities, as well as in skillful supervision and mentoring of your (or someone else's) students. All these things are considered in my department, and the potential to obtain research funding (and produce good quality PhD graduates) are both seen as very important. Having no real first author publications has given people problems before, as has having no collaborations or supervisory-capacity publications.
To address sole authorship: In engineering, I find (at least in my field) that sole authorship is uncommon, and have not seen it regarded differently than first authorship on a paper with several authors. As an academic you form part of a group and it is expected you'll work individually as well as part of a team. Therefore sole author doesn't seem to be treated differently to first author.
