Left-to-right, short-circuit evaluation of the logical operators is so deeply rooted in C idiom that a mainstream compiler that doesn't respect it would be practically unusable. But perhaps it wouldn't work in a hobbyist compiler, or a non-standard dialect of C.
While valid according to the language definition, assignments within conditional expressions do have a reputation for being error-prone and not always the easiest code to read. Sometimes they're still neater than the available alternatives.
However, IMHO in this case, it feels a bit too easy to break the code during later maintenance edits. For example, someone might simplify the code by removing the explicit comparisons with 0 that are unnecessary here, and then later someone else might "fix" an "obvious bug" by replacing the "(err = x) || (err = y)" construction with the common construction "(err == x) || (err == y)" that they assumed was intended, perhaps prompted by a compiler warning. Obviously they shouldn't if they didn't properly understand the code, but when has that ever been a reliable guarantee? :-)
If not, `|| (err = check())` is equivalent to separate checks which also also `goto fail` immediately.