No, it's definitely not "necessary".
I'm not an engineering major, but roomed with one, did a general engineering minor, and worked in/around mechanical, nuclear, mining and chemical engineering (had electrical on staff too). Passed my EIT and was at one time, about to take the PE (mechanical) exam. Most engineers in the workplace don't even use calculus, let alone epsilon-delta. For that matter I've worked with a (small number of, but still violates a Euclidean "necessary") licensed PEs who didn't even have a college degree.
In their normal undergrad training (fluids, thermo, etc.), engineers will often use calculus. So you need calculus to get through a BS in E. But you don't need or use epsilon-delta in those courses, either in the derivations or the homework drill or the projects.
Even if we use a less picky interpretation of "necessary" (very helpful), I would not put epsilon delta up there. For one thing, the current stereotypical STEM training in the US, gives an exposure to epsilon delta, with a few problems, at the beginning of a calc course. But doesn't come back to it, doesn't use it. So it's not even that important for undergrad calculus or diffyQs.
Now, does it hurt to have something like this? I don't think so. Especially given the limited time spent on in a typical calculus course based on a text like Thomas. I mean at least you've seen it so someone mentions it, it's not a mythical creature. And it's really just a bunch of detailed algebra pushing symbols and equations around (my same feeling about series)--maybe builds the algebra muscles in a manner helpful for and similar to multistep homework problems in fluid hydraulics or power systems. But it's not firmly connected--just general muscle building. For the vainshingly small but not converging to zero population of undergrad engineers that go on to some theoretical Ph.D. (and even within that a small set of them) and turn out to need this sort of stuff, they at least were exposed to it. And can pick it up more as needed, for their work. (Not to say they couldn't even with no previous exposure.)
Being practical, we can always come up with things that might in some circumstance be helpful (learning Latin, say). But life is finite. It is practical. It's an engineering problem, with constraints, costs. etc. So I definitely would not use the strong word necessary (even if we use it colloquially to mean strongly important) to describe formal limits. There are a gazillion things useful to engineers and a lot on their slate already. There is limited time in the day and limited brains in the skull. So, I wouldn't go beyond the week or two at beginning of a calc course, as done now.