One does not need to use programming whatsoever as an electronics engineer. However, you'll probably end up doing lots and lots of analog electronics or lots and lots of discrete logic.
I have a friend who wanted to make an LED flash when a button was pushed. He was going to use a 555 timer circuit along with a transistor (around a half-dozen parts, utilizing some resistors he didn't have on hand and probably an hour or two of tinkering). I suggested that he use a small microcontroller instead, and a half-dozen or so lines of c code. Or a few more lines to make really cool lighting effects (this was for a Deathstar cake his fiancée had been commissioned to make). Ironically, I'm the EE, and he's the CompSci.
In a nutshell, programming (and microcontrollers, CPLDs, FPGAs, etc.) allow you to trade hardware for software. And if you mess up, it's often much easier to fix software mistakes (by reprogramming, or uploading new software to reprogrammable devices) than hardware mistakes (track cutting, ugly board hacks, etc.)
And of course, not all EEs (nor even a significant portion of them) end up doing electronics design (e.g. power distribution systems), design at all (e.g. testing or technical writing), or even something that nebulously falls under the aegis of engineering! (E.g. management or sales, and that doesn't count the people that decide to ditch their EE degrees and go to law school or become a priest, or a banker--things that some of my classmates actually ended up doing!)